Motivation and emotion/Assessment/Chapter

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&lt;div style=&quot;text-align: center;&quot;&gt;''Collaborative online book chapter authoring''
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&lt;!-- Count down --&gt;
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{{countdown
 |year       = 2026
 |month      = 10
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--&gt;&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
{{/Contents/}}&lt;/div&gt;
{{TOCright}}

==Overview==
* Weight: 50%
* Due {{/Due}}
* Tasks
** Author an online [[Motivation and emotion/Book|book chapter]] (up to 4,000 words) that explains key psychological theory and research about a unique, specific motivation or emotion topic
** Develop the chapter by building on the [[Motivation and emotion/Assessment/Topic|topic development]] plan and addressing feedback
** Contribute to the development of other book chapters (social contribution)
* Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]]

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing marking rubric]
*Feedback will be provided to explain how well the chapter meets the marking criteria
*Marks and feedback will be provided when official university grades are released (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates])
**Marks will be available via {{Motivation and emotion/Canvas}}
**Written feedback will be available via the chapter's Wikiversity discussion page
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions

==Deferred exams==
* Apply for a deferred exam using the University's [https://www.canberra.edu.au/content/myuc/home/my-studies/exams/about-uc-exams.html deferred exam application] with appropriate documentary evidence
* Submissions are accepted up to 3 days late (-10% per day late)
* If you don't submit this assessment it is unlikely that you will pass the unit

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto; vertical-align:top;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| style=&quot;vertical-align:top;&quot; | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| style=&quot;vertical-align:top;&quot; | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| style=&quot;vertical-align:top;&quot; | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;4&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Review scholarly knowledge in an open, online environment and address feedback.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive
| style=&quot;vertical-align:top;&quot; | Produce an online book chapter about a novel motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Up-to-date knowledge and skills
| style=&quot;vertical-align:top;&quot; | Utilise the most relevant psychological theory and research to address a practical question.
|-
| style=&quot;vertical-align:top;&quot; | Solve problems via thinking
| style=&quot;vertical-align:top;&quot; | Use critical thinking to explain how psychological science can address real-world problems.
|-
| rowspan=&quot;3&quot; style=&quot;vertical-align:top;&quot; | '''Be a global citizen'''
| style=&quot;vertical-align:top;&quot; | Informed and balanced
| style=&quot;vertical-align:top;&quot; | Provide a balanced, critical chapter which is accessible to a lay audience.
|-
| style=&quot;vertical-align:top;&quot; | Communicate diversely
| style=&quot;vertical-align:top;&quot; | Collaborate with peers to communicate knowledge openly with a global audience.
|-
| style=&quot;vertical-align:top;&quot; | Creative use of technology
| style=&quot;vertical-align:top;&quot; | Learn how to collaborate using wiki technology.
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Engage in new ideas
| style=&quot;vertical-align:top;&quot; | Engage in a collaborative learning culture by incorporating feedback and suggestions.
|-
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Experience project work in a collaborative, online editing environment.
|}

==Instructions==
The following instructions should be used to guide the development of the book chapter.

===Theme===
* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

===Audience===
* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise.

===Wikiversity===
* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

===Topic===
* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

===Collaboration and feedback===
* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread&lt;!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)--&gt;)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

===Length (word count)===
{{Anchor|Wordcount}}{{Anchor|Word count}}
* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]]

===Submission===
* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]
Book chapters will be marked against the following criteria.

===Overview (5%)===
* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

===Theory (20%)===
* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

===Research (25%)===
* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* [[w:Critical thinking|Critical analyse]] the key research findings, including limitations and implications

===Integration (10%)===
* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

===Conclusion (5%)===
* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

===Style (20%)===
* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[/Readability|Readable]] for a layperson interested in psychological science
** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., &quot;it&quot;, &quot;they&quot;). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., &quot;I&quot; and &quot;we&quot;) and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

===Learning features (5%)===
* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &amp;lt;nowiki&amp;gt;[[#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter
{{anchor|Socialcontribution}}

{{anchor|Social contribution}}
===Social contribution (10%)===
* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk page]]s
** '''Media uploads''': Upload free-to-use educational images to [[commons:|Wikimedia Commons]]
** '''{{Motivation and emotion/Canvas}} discussion posts'''
* '''Evidence''':
** Provide a [[mw:Help:Lists|numbered list]] with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.&lt;!-- To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Summarise each edit on your user page (in a section called &quot;Social contributions&quot;) --&gt;
** More info: [[/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00
* '''Bonus marks'''
** Up to 5 bonus marks (out of 100 for the book chapter) will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
** Approximately 5% of students receive social contribution bonus marks.
** This could include extensive copyediting of multiple chapters and/or regular feedback or support on multiple chapter discussion pages. It may also involve substantial activity on the UCLearn discussion forum.

==Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1NUNjMCM5xOVsHj1LJuRnK8BgEdXPn4TbRf-Q6PH3oXg/edit?usp=sharing Marking rubric]}}

==Examples==
Examples of high quality [[Motivation and emotion/Book|motivation and emotion book chapters]]:
* [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022)
* [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016)
* [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019)
* [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019)

Note that as of 2025, chapters no longer include multimedia presentations.

For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]&lt;!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] --&gt;.

==Licensing==
Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener.

==See also==
* [[/FAQ/]]
* [[Motivation and emotion/Book|Previous chapters]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]]
** [[Template:MEBF|Feedback template]]
** [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing Marking rubric]
* [[#Socialcontribution|Social contributions]]
** [[/Search for chapters to improve/]]
** [[/Summarising social contributions/]]
* [[Motivation and emotion/Tutorials|Tutorials]]
&lt;!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --&gt;
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]]
** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]]
* Wikiversity
** [[/Feature boxes/]]
** [[/Figures/]]
** [[How to find free-to-use images|Find free images]]
** [[/Tables/]]
* [[/Writing tips/]]
** [[/How to handle a lack of information/|Handling a lack of information]]
** [[/Readability|Readability]]
** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
** [[/Word count|Word count]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Chapter| ]]
[[Category:Motivation and emotion guidelines]]</text>
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&lt;div style=&quot;text-align: center;&quot;&gt;''Collaborative online book chapter authoring''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
{{countdown
 |year       = 2026
 |month      = 10
 |day        = 12
 |hour       = 0
 |event = this assessment is due
}}
--&gt;&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
{{/Contents/}}&lt;/div&gt;
{{TOCright}}

==Overview==
* Weight: 50%
* Due {{/Due}}
* Tasks
** Author an online [[Motivation and emotion/Book|book chapter]] (up to 4,000 words) that explains key psychological theory and research about a unique, specific motivation or emotion topic
** Develop the chapter by building on the [[Motivation and emotion/Assessment/Topic|topic development]] plan and addressing feedback
** Contribute to the development of other book chapters (social contribution)
* Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]]

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing marking rubric]
* Marks will be provided via {{Motivation and emotion/Canvas}} when the university releases unit results (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates])
Up to 5 bonus marks will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level
Written feedback about how well the chapter addresses the marking criteria will be available via the topic's Wikiversity discussion page
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions

==Deferred exams==
* Apply for a deferred exam using the University's [https://www.canberra.edu.au/content/myuc/home/my-studies/exams/about-uc-exams.html deferred exam application] with appropriate documentary evidence
* Submissions are accepted up to 3 days late (-10% per day late)
* If you don't submit this assessment it is unlikely that you will pass the unit

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto; vertical-align:top;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| style=&quot;vertical-align:top;&quot; | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| style=&quot;vertical-align:top;&quot; | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| style=&quot;vertical-align:top;&quot; | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;4&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Review scholarly knowledge in an open, online environment and address feedback.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive
| style=&quot;vertical-align:top;&quot; | Produce an online book chapter about a novel motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Up-to-date knowledge and skills
| style=&quot;vertical-align:top;&quot; | Utilise the most relevant psychological theory and research to address a practical question.
|-
| style=&quot;vertical-align:top;&quot; | Solve problems via thinking
| style=&quot;vertical-align:top;&quot; | Use critical thinking to explain how psychological science can address real-world problems.
|-
| rowspan=&quot;3&quot; style=&quot;vertical-align:top;&quot; | '''Be a global citizen'''
| style=&quot;vertical-align:top;&quot; | Informed and balanced
| style=&quot;vertical-align:top;&quot; | Provide a balanced, critical chapter which is accessible to a lay audience.
|-
| style=&quot;vertical-align:top;&quot; | Communicate diversely
| style=&quot;vertical-align:top;&quot; | Collaborate with peers to communicate knowledge openly with a global audience.
|-
| style=&quot;vertical-align:top;&quot; | Creative use of technology
| style=&quot;vertical-align:top;&quot; | Learn how to collaborate using wiki technology.
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Engage in new ideas
| style=&quot;vertical-align:top;&quot; | Engage in a collaborative learning culture by incorporating feedback and suggestions.
|-
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Experience project work in a collaborative, online editing environment.
|}

==Instructions==
The following instructions should be used to guide the development of the book chapter.

===Theme===
* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

===Audience===
* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise.

===Wikiversity===
* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

===Topic===
* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

===Collaboration and feedback===
* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread&lt;!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)--&gt;)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

===Length (word count)===
{{Anchor|Wordcount}}{{Anchor|Word count}}
* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]]

===Submission===
* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]
Book chapters will be marked against the following criteria.

===Overview (5%)===
* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

===Theory (20%)===
* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

===Research (25%)===
* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* [[w:Critical thinking|Critical analyse]] the key research findings, including limitations and implications

===Integration (10%)===
* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

===Conclusion (5%)===
* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

===Style (20%)===
* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[/Readability|Readable]] for a layperson interested in psychological science
** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., &quot;it&quot;, &quot;they&quot;). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., &quot;I&quot; and &quot;we&quot;) and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

===Learning features (5%)===
* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &amp;lt;nowiki&amp;gt;[[#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter
{{anchor|Socialcontribution}}

{{anchor|Social contribution}}
===Social contribution (10%)===
* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk page]]s
** '''Media uploads''': Upload free-to-use educational images to [[commons:|Wikimedia Commons]]
** '''{{Motivation and emotion/Canvas}} discussion posts'''
* '''Evidence''':
** Provide a [[mw:Help:Lists|numbered list]] with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.&lt;!-- To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Summarise each edit on your user page (in a section called &quot;Social contributions&quot;) --&gt;
** More info: [[/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00
* '''Bonus marks'''
** Up to 5 bonus marks (out of 100 for the book chapter) will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
** Approximately 5% of students receive social contribution bonus marks.
** This could include extensive copyediting of multiple chapters and/or regular feedback or support on multiple chapter discussion pages. It may also involve substantial activity on the UCLearn discussion forum.

==Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1NUNjMCM5xOVsHj1LJuRnK8BgEdXPn4TbRf-Q6PH3oXg/edit?usp=sharing Marking rubric]}}

==Examples==
Examples of high quality [[Motivation and emotion/Book|motivation and emotion book chapters]]:
* [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022)
* [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016)
* [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019)
* [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019)

Note that as of 2025, chapters no longer include multimedia presentations.

For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]&lt;!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] --&gt;.

==Licensing==
Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener.

==See also==
* [[/FAQ/]]
* [[Motivation and emotion/Book|Previous chapters]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]]
** [[Template:MEBF|Feedback template]]
** [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing Marking rubric]
* [[#Socialcontribution|Social contributions]]
** [[/Search for chapters to improve/]]
** [[/Summarising social contributions/]]
* [[Motivation and emotion/Tutorials|Tutorials]]
&lt;!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --&gt;
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]]
** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]]
* Wikiversity
** [[/Feature boxes/]]
** [[/Figures/]]
** [[How to find free-to-use images|Find free images]]
** [[/Tables/]]
* [[/Writing tips/]]
** [[/How to handle a lack of information/|Handling a lack of information]]
** [[/Readability|Readability]]
** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
** [[/Word count|Word count]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Chapter| ]]
[[Category:Motivation and emotion guidelines]]</text>
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&lt;div style=&quot;text-align: center;&quot;&gt;''Collaborative online book chapter authoring''
&lt;!-- ---------------------------------- ---&gt;
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{{/Contents/}}&lt;/div&gt;
{{TOCright}}

==Overview==
* Weight: 50%
* Due {{/Due}}
* Tasks
** Author an online [[Motivation and emotion/Book|book chapter]] (up to 4,000 words) that explains key psychological theory and research about a unique, specific motivation or emotion topic
** Develop the chapter by building on the [[Motivation and emotion/Assessment/Topic|topic development]] plan and addressing feedback
** Contribute to the development of other book chapters (social contribution)
* Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]]

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing marking rubric]
* Marks will be provided via {{Motivation and emotion/Canvas}} when the university releases unit results (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates])
* Up to 5 bonus marks will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level
* Written feedback about how well the chapter addresses the marking criteria will be available via the topic's Wikiversity discussion page
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions

==Deferred exams==
* Apply for a deferred exam using the University's [https://www.canberra.edu.au/content/myuc/home/my-studies/exams/about-uc-exams.html deferred exam application] with appropriate documentary evidence
* Submissions are accepted up to 3 days late (-10% per day late)
* If you don't submit this assessment it is unlikely that you will pass the unit

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto; vertical-align:top;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| style=&quot;vertical-align:top;&quot; | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| style=&quot;vertical-align:top;&quot; | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| style=&quot;vertical-align:top;&quot; | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;4&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Review scholarly knowledge in an open, online environment and address feedback.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive
| style=&quot;vertical-align:top;&quot; | Produce an online book chapter about a novel motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Up-to-date knowledge and skills
| style=&quot;vertical-align:top;&quot; | Utilise the most relevant psychological theory and research to address a practical question.
|-
| style=&quot;vertical-align:top;&quot; | Solve problems via thinking
| style=&quot;vertical-align:top;&quot; | Use critical thinking to explain how psychological science can address real-world problems.
|-
| rowspan=&quot;3&quot; style=&quot;vertical-align:top;&quot; | '''Be a global citizen'''
| style=&quot;vertical-align:top;&quot; | Informed and balanced
| style=&quot;vertical-align:top;&quot; | Provide a balanced, critical chapter which is accessible to a lay audience.
|-
| style=&quot;vertical-align:top;&quot; | Communicate diversely
| style=&quot;vertical-align:top;&quot; | Collaborate with peers to communicate knowledge openly with a global audience.
|-
| style=&quot;vertical-align:top;&quot; | Creative use of technology
| style=&quot;vertical-align:top;&quot; | Learn how to collaborate using wiki technology.
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Engage in new ideas
| style=&quot;vertical-align:top;&quot; | Engage in a collaborative learning culture by incorporating feedback and suggestions.
|-
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Experience project work in a collaborative, online editing environment.
|}

==Instructions==
The following instructions should be used to guide the development of the book chapter.

===Theme===
* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

===Audience===
* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise.

===Wikiversity===
* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

===Topic===
* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

===Collaboration and feedback===
* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread&lt;!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)--&gt;)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

===Length (word count)===
{{Anchor|Wordcount}}{{Anchor|Word count}}
* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]]

===Submission===
* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]
Book chapters will be marked against the following criteria.

===Overview (5%)===
* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

===Theory (20%)===
* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

===Research (25%)===
* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* [[w:Critical thinking|Critical analyse]] the key research findings, including limitations and implications

===Integration (10%)===
* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

===Conclusion (5%)===
* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

===Style (20%)===
* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[/Readability|Readable]] for a layperson interested in psychological science
** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., &quot;it&quot;, &quot;they&quot;). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., &quot;I&quot; and &quot;we&quot;) and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

===Learning features (5%)===
* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &amp;lt;nowiki&amp;gt;[[#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter
{{anchor|Socialcontribution}}

{{anchor|Social contribution}}
===Social contribution (10%)===
* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk page]]s
** '''Media uploads''': Upload free-to-use educational images to [[commons:|Wikimedia Commons]]
** '''{{Motivation and emotion/Canvas}} discussion posts'''
* '''Evidence''':
** Provide a [[mw:Help:Lists|numbered list]] with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.&lt;!-- To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Summarise each edit on your user page (in a section called &quot;Social contributions&quot;) --&gt;
** More info: [[/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00
* '''Bonus marks'''
** Up to 5 bonus marks (out of 100 for the book chapter) will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
** Approximately 5% of students receive social contribution bonus marks.
** This could include extensive copyediting of multiple chapters and/or regular feedback or support on multiple chapter discussion pages. It may also involve substantial activity on the UCLearn discussion forum.

==Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1NUNjMCM5xOVsHj1LJuRnK8BgEdXPn4TbRf-Q6PH3oXg/edit?usp=sharing Marking rubric]}}

==Examples==
Examples of high quality [[Motivation and emotion/Book|motivation and emotion book chapters]]:
* [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022)
* [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016)
* [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019)
* [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019)

Note that as of 2025, chapters no longer include multimedia presentations.

For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]&lt;!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] --&gt;.

==Licensing==
Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener.

==See also==
* [[/FAQ/]]
* [[Motivation and emotion/Book|Previous chapters]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]]
** [[Template:MEBF|Feedback template]]
** [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing Marking rubric]
* [[#Socialcontribution|Social contributions]]
** [[/Search for chapters to improve/]]
** [[/Summarising social contributions/]]
* [[Motivation and emotion/Tutorials|Tutorials]]
&lt;!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --&gt;
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]]
** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]]
* Wikiversity
** [[/Feature boxes/]]
** [[/Figures/]]
** [[How to find free-to-use images|Find free images]]
** [[/Tables/]]
* [[/Writing tips/]]
** [[/How to handle a lack of information/|Handling a lack of information]]
** [[/Readability|Readability]]
** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
** [[/Word count|Word count]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Chapter| ]]
[[Category:Motivation and emotion guidelines]]</text>
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&lt;div style=&quot;text-align: center;&quot;&gt;''Collaborative online book chapter authoring''
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==Overview==
* Weight: 50%
* Due {{/Due}}
* Tasks
** Author an online [[Motivation and emotion/Book|book chapter]] (up to 4,000 words) that explains key psychological theory and research about a unique, specific motivation or emotion topic.
** Develop the chapter by building on the [[Motivation and emotion/Assessment/Topic|topic development]] plan and addressing feedback.
** Contribute to the development of other book chapters (social contribution).
* Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing marking rubric]
* Marks will be provided via {{Motivation and emotion/Canvas}} when the university releases unit results (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates])
* Up to 5 bonus marks will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level
* Written feedback about how well the chapter addresses the marking criteria will be available via the topic's Wikiversity discussion page
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions

==Deferred exams==
* Apply for a deferred exam using the University's [https://www.canberra.edu.au/content/myuc/home/my-studies/exams/about-uc-exams.html deferred exam application] with appropriate documentary evidence
* Submissions are accepted up to 3 days late (-10% per day late)
* If you don't submit this assessment it is unlikely that you will pass the unit

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto; vertical-align:top;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| style=&quot;vertical-align:top;&quot; | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| style=&quot;vertical-align:top;&quot; | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| style=&quot;vertical-align:top;&quot; | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;4&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Review scholarly knowledge in an open, online environment and address feedback.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive
| style=&quot;vertical-align:top;&quot; | Produce an online book chapter about a novel motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Up-to-date knowledge and skills
| style=&quot;vertical-align:top;&quot; | Utilise the most relevant psychological theory and research to address a practical question.
|-
| style=&quot;vertical-align:top;&quot; | Solve problems via thinking
| style=&quot;vertical-align:top;&quot; | Use critical thinking to explain how psychological science can address real-world problems.
|-
| rowspan=&quot;3&quot; style=&quot;vertical-align:top;&quot; | '''Be a global citizen'''
| style=&quot;vertical-align:top;&quot; | Informed and balanced
| style=&quot;vertical-align:top;&quot; | Provide a balanced, critical chapter which is accessible to a lay audience.
|-
| style=&quot;vertical-align:top;&quot; | Communicate diversely
| style=&quot;vertical-align:top;&quot; | Collaborate with peers to communicate knowledge openly with a global audience.
|-
| style=&quot;vertical-align:top;&quot; | Creative use of technology
| style=&quot;vertical-align:top;&quot; | Learn how to collaborate using wiki technology.
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Engage in new ideas
| style=&quot;vertical-align:top;&quot; | Engage in a collaborative learning culture by incorporating feedback and suggestions.
|-
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Experience project work in a collaborative, online editing environment.
|}

==Instructions==
The following instructions should be used to guide the development of the book chapter.

===Theme===
* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

===Audience===
* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise.

===Wikiversity===
* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

===Topic===
* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

===Collaboration and feedback===
* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread&lt;!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)--&gt;)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

===Length (word count)===
{{Anchor|Wordcount}}{{Anchor|Word count}}
* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]]

===Submission===
* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]
Book chapters will be marked against the following criteria.

===Overview (5%)===
* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

===Theory (20%)===
* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

===Research (25%)===
* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* [[w:Critical thinking|Critical analyse]] the key research findings, including limitations and implications

===Integration (10%)===
* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

===Conclusion (5%)===
* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

===Style (20%)===
* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[/Readability|Readable]] for a layperson interested in psychological science
** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., &quot;it&quot;, &quot;they&quot;). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., &quot;I&quot; and &quot;we&quot;) and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

===Learning features (5%)===
* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &amp;lt;nowiki&amp;gt;[[#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter
{{anchor|Socialcontribution}}

{{anchor|Social contribution}}
===Social contribution (10%)===
* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk page]]s
** '''Media uploads''': Upload free-to-use educational images to [[commons:|Wikimedia Commons]]
** '''{{Motivation and emotion/Canvas}} discussion posts'''
* '''Evidence''':
** Provide a [[mw:Help:Lists|numbered list]] with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.&lt;!-- To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Summarise each edit on your user page (in a section called &quot;Social contributions&quot;) --&gt;
** More info: [[/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00
* '''Bonus marks'''
** Up to 5 bonus marks (out of 100 for the book chapter) will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
** Approximately 5% of students receive social contribution bonus marks.
** This could include extensive copyediting of multiple chapters and/or regular feedback or support on multiple chapter discussion pages. It may also involve substantial activity on the UCLearn discussion forum.

==Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1NUNjMCM5xOVsHj1LJuRnK8BgEdXPn4TbRf-Q6PH3oXg/edit?usp=sharing Marking rubric]}}

==Examples==
Examples of high quality [[Motivation and emotion/Book|motivation and emotion book chapters]]:
* [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022)
* [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016)
* [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019)
* [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019)

Note that as of 2025, chapters no longer include multimedia presentations.

For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]&lt;!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] --&gt;.

==Licensing==
Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener.

==See also==
* [[/FAQ/]]
* [[Motivation and emotion/Book|Previous chapters]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]]
** [[Template:MEBF|Feedback template]]
** [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing Marking rubric]
* [[#Socialcontribution|Social contributions]]
** [[/Search for chapters to improve/]]
** [[/Summarising social contributions/]]
* [[Motivation and emotion/Tutorials|Tutorials]]
&lt;!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --&gt;
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]]
** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]]
* Wikiversity
** [[/Feature boxes/]]
** [[/Figures/]]
** [[How to find free-to-use images|Find free images]]
** [[/Tables/]]
* [[/Writing tips/]]
** [[/How to handle a lack of information/|Handling a lack of information]]
** [[/Readability|Readability]]
** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
** [[/Word count|Word count]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Chapter| ]]
[[Category:Motivation and emotion guidelines]]</text>
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      <text bytes="21857" sha1="clbcw98xln6ubp6nt4vjnvggws6v1vt" xml:space="preserve">{{title|Book chapter — Guidelines}}
&lt;div style=&quot;text-align: center;&quot;&gt;''Collaborative online book chapter authoring''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
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--&gt;&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
{{/Contents/}}&lt;/div&gt;
{{TOCright}}

==Overview==
* Weight: 50%
* Due {{/Due}}
* Tasks
** Author an online [[Motivation and emotion/Book|book chapter]] (up to 4,000 words) that explains key psychological theory and research about a unique, specific motivation or emotion topic.
** Develop the chapter by building on the [[Motivation and emotion/Assessment/Topic|topic development]] plan and addressing feedback.
** Contribute to the development of other book chapters (social contribution).
* Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].

==Marking and feedback==
* Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing marking rubric].
* Marks will be provided via {{Motivation and emotion/Canvas}} when the university releases unit results (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates]).
* Up to 5 bonus marks will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
* Written feedback about how well the chapter addresses the marking criteria will be available via the topic's Wikiversity discussion page.
* Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions.

==Deferred exams==
* Apply for a deferred exam using the University's [https://www.canberra.edu.au/content/myuc/home/my-studies/exams/about-uc-exams.html deferred exam application] with appropriate documentary evidence
* Submissions are accepted up to 3 days late (-10% per day late)
* If you don't submit this assessment it is unlikely that you will pass the unit

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto; vertical-align:top;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| style=&quot;vertical-align:top;&quot; | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| style=&quot;vertical-align:top;&quot; | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| style=&quot;vertical-align:top;&quot; | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;4&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Review scholarly knowledge in an open, online environment and address feedback.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive
| style=&quot;vertical-align:top;&quot; | Produce an online book chapter about a novel motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Up-to-date knowledge and skills
| style=&quot;vertical-align:top;&quot; | Utilise the most relevant psychological theory and research to address a practical question.
|-
| style=&quot;vertical-align:top;&quot; | Solve problems via thinking
| style=&quot;vertical-align:top;&quot; | Use critical thinking to explain how psychological science can address real-world problems.
|-
| rowspan=&quot;3&quot; style=&quot;vertical-align:top;&quot; | '''Be a global citizen'''
| style=&quot;vertical-align:top;&quot; | Informed and balanced
| style=&quot;vertical-align:top;&quot; | Provide a balanced, critical chapter which is accessible to a lay audience.
|-
| style=&quot;vertical-align:top;&quot; | Communicate diversely
| style=&quot;vertical-align:top;&quot; | Collaborate with peers to communicate knowledge openly with a global audience.
|-
| style=&quot;vertical-align:top;&quot; | Creative use of technology
| style=&quot;vertical-align:top;&quot; | Learn how to collaborate using wiki technology.
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Engage in new ideas
| style=&quot;vertical-align:top;&quot; | Engage in a collaborative learning culture by incorporating feedback and suggestions.
|-
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Experience project work in a collaborative, online editing environment.
|}

==Instructions==
The following instructions should be used to guide the development of the book chapter.

===Theme===
* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

===Audience===
* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise.

===Wikiversity===
* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

===Topic===
* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

===Collaboration and feedback===
* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread&lt;!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)--&gt;)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

===Length (word count)===
{{Anchor|Wordcount}}{{Anchor|Word count}}
* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]]

===Submission===
* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]
Book chapters will be marked against the following criteria.

===Overview (5%)===
* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

===Theory (20%)===
* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

===Research (25%)===
* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* [[w:Critical thinking|Critical analyse]] the key research findings, including limitations and implications

===Integration (10%)===
* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

===Conclusion (5%)===
* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

===Style (20%)===
* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[/Readability|Readable]] for a layperson interested in psychological science
** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., &quot;it&quot;, &quot;they&quot;). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., &quot;I&quot; and &quot;we&quot;) and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

===Learning features (5%)===
* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &amp;lt;nowiki&amp;gt;[[#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter
{{anchor|Socialcontribution}}

{{anchor|Social contribution}}
===Social contribution (10%)===
* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk page]]s
** '''Media uploads''': Upload free-to-use educational images to [[commons:|Wikimedia Commons]]
** '''{{Motivation and emotion/Canvas}} discussion posts'''
* '''Evidence''':
** Provide a [[mw:Help:Lists|numbered list]] with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.&lt;!-- To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Summarise each edit on your user page (in a section called &quot;Social contributions&quot;) --&gt;
** More info: [[/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00
* '''Bonus marks'''
** Up to 5 bonus marks (out of 100 for the book chapter) will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
** Approximately 5% of students receive social contribution bonus marks.
** This could include extensive copyediting of multiple chapters and/or regular feedback or support on multiple chapter discussion pages. It may also involve substantial activity on the UCLearn discussion forum.

==Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1NUNjMCM5xOVsHj1LJuRnK8BgEdXPn4TbRf-Q6PH3oXg/edit?usp=sharing Marking rubric]}}

==Examples==
Examples of high quality [[Motivation and emotion/Book|motivation and emotion book chapters]]:
* [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022)
* [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016)
* [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019)
* [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019)

Note that as of 2025, chapters no longer include multimedia presentations.

For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]&lt;!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] --&gt;.

==Licensing==
Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener.

==See also==
* [[/FAQ/]]
* [[Motivation and emotion/Book|Previous chapters]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]]
** [[Template:MEBF|Feedback template]]
** [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing Marking rubric]
* [[#Socialcontribution|Social contributions]]
** [[/Search for chapters to improve/]]
** [[/Summarising social contributions/]]
* [[Motivation and emotion/Tutorials|Tutorials]]
&lt;!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --&gt;
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]]
** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]]
* Wikiversity
** [[/Feature boxes/]]
** [[/Figures/]]
** [[How to find free-to-use images|Find free images]]
** [[/Tables/]]
* [[/Writing tips/]]
** [[/How to handle a lack of information/|Handling a lack of information]]
** [[/Readability|Readability]]
** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
** [[/Word count|Word count]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Chapter| ]]
[[Category:Motivation and emotion guidelines]]</text>
      <sha1>clbcw98xln6ubp6nt4vjnvggws6v1vt</sha1>
    </revision>
    <revision>
      <id>2819224</id>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
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&lt;div style=&quot;text-align: center;&quot;&gt;''Collaborative online book chapter authoring''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
{{countdown
 |year       = 2026
 |month      = 10
 |day        = 12
 |hour       = 0
 |event = this assessment is due
}}
--&gt;&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
{{/Contents/}}&lt;/div&gt;
{{TOCright}}

==Overview==
* Weight: 50%
* Due {{/Due}}
* Tasks
** Author an online [[Motivation and emotion/Book|book chapter]] (up to 4,000 words) that explains key psychological theory and research about a unique, specific motivation or emotion topic.
** Develop the chapter by building on the [[Motivation and emotion/Assessment/Topic|topic development]] plan and addressing feedback.
** Contribute to the development of other book chapters (social contribution).
* Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].

==Marking and feedback==
* Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing marking rubric].
* Marks will be provided via {{Motivation and emotion/Canvas}} when the university releases unit results (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates]).
* Up to 5 bonus marks will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
* Written feedback about how well the chapter addresses the marking criteria will be available via the topic's Wikiversity discussion page.
* Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions.

==Extensions and late submissions==
* Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence.
* Submissions will be accepted up to 3 days late (-10% per day).
* If you don't submit this assessment it is unlikely that you will pass the unit.

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto; vertical-align:top;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| style=&quot;vertical-align:top;&quot; | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| style=&quot;vertical-align:top;&quot; | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| style=&quot;vertical-align:top;&quot; | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;4&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Review scholarly knowledge in an open, online environment and address feedback.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive
| style=&quot;vertical-align:top;&quot; | Produce an online book chapter about a novel motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Up-to-date knowledge and skills
| style=&quot;vertical-align:top;&quot; | Utilise the most relevant psychological theory and research to address a practical question.
|-
| style=&quot;vertical-align:top;&quot; | Solve problems via thinking
| style=&quot;vertical-align:top;&quot; | Use critical thinking to explain how psychological science can address real-world problems.
|-
| rowspan=&quot;3&quot; style=&quot;vertical-align:top;&quot; | '''Be a global citizen'''
| style=&quot;vertical-align:top;&quot; | Informed and balanced
| style=&quot;vertical-align:top;&quot; | Provide a balanced, critical chapter which is accessible to a lay audience.
|-
| style=&quot;vertical-align:top;&quot; | Communicate diversely
| style=&quot;vertical-align:top;&quot; | Collaborate with peers to communicate knowledge openly with a global audience.
|-
| style=&quot;vertical-align:top;&quot; | Creative use of technology
| style=&quot;vertical-align:top;&quot; | Learn how to collaborate using wiki technology.
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Engage in new ideas
| style=&quot;vertical-align:top;&quot; | Engage in a collaborative learning culture by incorporating feedback and suggestions.
|-
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Experience project work in a collaborative, online editing environment.
|}

==Instructions==
The following instructions should be used to guide the development of the book chapter.

===Theme===
* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

===Audience===
* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise.

===Wikiversity===
* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

===Topic===
* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

===Collaboration and feedback===
* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread&lt;!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)--&gt;)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

===Length (word count)===
{{Anchor|Wordcount}}{{Anchor|Word count}}
* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]]

===Submission===
* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]
Book chapters will be marked against the following criteria.

===Overview (5%)===
* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

===Theory (20%)===
* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

===Research (25%)===
* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* [[w:Critical thinking|Critical analyse]] the key research findings, including limitations and implications

===Integration (10%)===
* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

===Conclusion (5%)===
* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

===Style (20%)===
* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[/Readability|Readable]] for a layperson interested in psychological science
** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., &quot;it&quot;, &quot;they&quot;). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., &quot;I&quot; and &quot;we&quot;) and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

===Learning features (5%)===
* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &amp;lt;nowiki&amp;gt;[[#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter
{{anchor|Socialcontribution}}

{{anchor|Social contribution}}
===Social contribution (10%)===
* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk page]]s
** '''Media uploads''': Upload free-to-use educational images to [[commons:|Wikimedia Commons]]
** '''{{Motivation and emotion/Canvas}} discussion posts'''
* '''Evidence''':
** Provide a [[mw:Help:Lists|numbered list]] with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.&lt;!-- To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Summarise each edit on your user page (in a section called &quot;Social contributions&quot;) --&gt;
** More info: [[/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00
* '''Bonus marks'''
** Up to 5 bonus marks (out of 100 for the book chapter) will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
** Approximately 5% of students receive social contribution bonus marks.
** This could include extensive copyediting of multiple chapters and/or regular feedback or support on multiple chapter discussion pages. It may also involve substantial activity on the UCLearn discussion forum.

==Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1NUNjMCM5xOVsHj1LJuRnK8BgEdXPn4TbRf-Q6PH3oXg/edit?usp=sharing Marking rubric]}}

==Examples==
Examples of high quality [[Motivation and emotion/Book|motivation and emotion book chapters]]:
* [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022)
* [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016)
* [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019)
* [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019)

Note that as of 2025, chapters no longer include multimedia presentations.

For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]&lt;!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] --&gt;.

==Licensing==
Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener.

==See also==
* [[/FAQ/]]
* [[Motivation and emotion/Book|Previous chapters]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]]
** [[Template:MEBF|Feedback template]]
** [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing Marking rubric]
* [[#Socialcontribution|Social contributions]]
** [[/Search for chapters to improve/]]
** [[/Summarising social contributions/]]
* [[Motivation and emotion/Tutorials|Tutorials]]
&lt;!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --&gt;
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]]
** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]]
* Wikiversity
** [[/Feature boxes/]]
** [[/Figures/]]
** [[How to find free-to-use images|Find free images]]
** [[/Tables/]]
* [[/Writing tips/]]
** [[/How to handle a lack of information/|Handling a lack of information]]
** [[/Readability|Readability]]
** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
** [[/Word count|Word count]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Chapter| ]]
[[Category:Motivation and emotion guidelines]]</text>
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      <text bytes="21939" sha1="m2d9fjrymjoslmkgkapviwr88y8t46v" xml:space="preserve">{{title|Book chapter — Guidelines}}
&lt;div style=&quot;text-align: center;&quot;&gt;''Collaborative online book chapter authoring''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
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--&gt;&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
{{/Contents/}}&lt;/div&gt;
{{TOCright}}

==Overview==
* Weight: 50%
* Due {{/Due}}
* Tasks
** Author an online [[Motivation and emotion/Book|book chapter]] (up to 4,000 words) that explains key psychological theory and research about a unique, specific motivation or emotion topic.
** Develop the chapter by building on the [[Motivation and emotion/Assessment/Topic|topic development]] plan and addressing feedback.
** Contribute to the development of other book chapters (social contribution).
* Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].

==Marking and feedback==
* Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing marking rubric].
* Marks will be provided via {{Motivation and emotion/Canvas}} when the university releases unit results (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates]).
* Up to 5 bonus marks will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level (see [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing rubric]).
* Written feedback about how well the chapter addresses the marking criteria will be available via the topic's Wikiversity discussion page.
* Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions.

==Extensions and late submissions==
* Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence.
* Submissions will be accepted up to 3 days late (-10% per day).
* If you don't submit this assessment it is unlikely that you will pass the unit.

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto; vertical-align:top;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| style=&quot;vertical-align:top;&quot; | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| style=&quot;vertical-align:top;&quot; | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| style=&quot;vertical-align:top;&quot; | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;4&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Review scholarly knowledge in an open, online environment and address feedback.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive
| style=&quot;vertical-align:top;&quot; | Produce an online book chapter about a novel motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Up-to-date knowledge and skills
| style=&quot;vertical-align:top;&quot; | Utilise the most relevant psychological theory and research to address a practical question.
|-
| style=&quot;vertical-align:top;&quot; | Solve problems via thinking
| style=&quot;vertical-align:top;&quot; | Use critical thinking to explain how psychological science can address real-world problems.
|-
| rowspan=&quot;3&quot; style=&quot;vertical-align:top;&quot; | '''Be a global citizen'''
| style=&quot;vertical-align:top;&quot; | Informed and balanced
| style=&quot;vertical-align:top;&quot; | Provide a balanced, critical chapter which is accessible to a lay audience.
|-
| style=&quot;vertical-align:top;&quot; | Communicate diversely
| style=&quot;vertical-align:top;&quot; | Collaborate with peers to communicate knowledge openly with a global audience.
|-
| style=&quot;vertical-align:top;&quot; | Creative use of technology
| style=&quot;vertical-align:top;&quot; | Learn how to collaborate using wiki technology.
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Engage in new ideas
| style=&quot;vertical-align:top;&quot; | Engage in a collaborative learning culture by incorporating feedback and suggestions.
|-
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Experience project work in a collaborative, online editing environment.
|}

==Instructions==
The following instructions should be used to guide the development of the book chapter.

===Theme===
* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

===Audience===
* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise.

===Wikiversity===
* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

===Topic===
* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

===Collaboration and feedback===
* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread&lt;!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)--&gt;)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

===Length (word count)===
{{Anchor|Wordcount}}{{Anchor|Word count}}
* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]]

===Submission===
* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]
Book chapters will be marked against the following criteria.

===Overview (5%)===
* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

===Theory (20%)===
* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

===Research (25%)===
* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* [[w:Critical thinking|Critical analyse]] the key research findings, including limitations and implications

===Integration (10%)===
* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

===Conclusion (5%)===
* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

===Style (20%)===
* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[/Readability|Readable]] for a layperson interested in psychological science
** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., &quot;it&quot;, &quot;they&quot;). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., &quot;I&quot; and &quot;we&quot;) and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

===Learning features (5%)===
* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &amp;lt;nowiki&amp;gt;[[#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter
{{anchor|Socialcontribution}}

{{anchor|Social contribution}}
===Social contribution (10%)===
* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk page]]s
** '''Media uploads''': Upload free-to-use educational images to [[commons:|Wikimedia Commons]]
** '''{{Motivation and emotion/Canvas}} discussion posts'''
* '''Evidence''':
** Provide a [[mw:Help:Lists|numbered list]] with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.&lt;!-- To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Summarise each edit on your user page (in a section called &quot;Social contributions&quot;) --&gt;
** More info: [[/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00
* '''Bonus marks'''
** Up to 5 bonus marks (out of 100 for the book chapter) will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
** Approximately 5% of students receive social contribution bonus marks.
** This could include extensive copyediting of multiple chapters and/or regular feedback or support on multiple chapter discussion pages. It may also involve substantial activity on the UCLearn discussion forum.

==Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1NUNjMCM5xOVsHj1LJuRnK8BgEdXPn4TbRf-Q6PH3oXg/edit?usp=sharing Marking rubric]}}

==Examples==
Examples of high quality [[Motivation and emotion/Book|motivation and emotion book chapters]]:
* [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022)
* [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016)
* [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019)
* [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019)

Note that as of 2025, chapters no longer include multimedia presentations.

For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]&lt;!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] --&gt;.

==Licensing==
Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener.

==See also==
* [[/FAQ/]]
* [[Motivation and emotion/Book|Previous chapters]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]]
** [[Template:MEBF|Feedback template]]
** [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing Marking rubric]
* [[#Socialcontribution|Social contributions]]
** [[/Search for chapters to improve/]]
** [[/Summarising social contributions/]]
* [[Motivation and emotion/Tutorials|Tutorials]]
&lt;!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --&gt;
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]]
** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]]
* Wikiversity
** [[/Feature boxes/]]
** [[/Figures/]]
** [[How to find free-to-use images|Find free images]]
** [[/Tables/]]
* [[/Writing tips/]]
** [[/How to handle a lack of information/|Handling a lack of information]]
** [[/Readability|Readability]]
** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
** [[/Word count|Word count]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Chapter| ]]
[[Category:Motivation and emotion guidelines]]</text>
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&lt;div style=&quot;text-align: center;&quot;&gt;''Collaborative online book chapter authoring''
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==Overview==
* Weight: 50%
* Due {{/Due}}
* Tasks
** Author an online [[Motivation and emotion/Book|book chapter]] (up to 4,000 words) that explains key psychological theory and research about a unique, specific motivation or emotion topic.
** Develop the chapter by building on the [[Motivation and emotion/Assessment/Topic|topic development]] plan and addressing feedback.
** Contribute to the development of other book chapters (social contribution).
* Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].

==Marking and feedback==
* Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing marking rubric].
* Marks will be provided via {{Motivation and emotion/Canvas}} when the university releases unit results (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates]).
* Up to 5 bonus marks will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level (see [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60 rubric]).
* Written feedback about how well the chapter addresses the marking criteria will be available via the topic's Wikiversity discussion page.
* Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions.

==Extensions and late submissions==
* Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence.
* Submissions will be accepted up to 3 days late (-10% per day).
* If you don't submit this assessment it is unlikely that you will pass the unit.

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto; vertical-align:top;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| style=&quot;vertical-align:top;&quot; | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| style=&quot;vertical-align:top;&quot; | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| style=&quot;vertical-align:top;&quot; | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;4&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Review scholarly knowledge in an open, online environment and address feedback.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive
| style=&quot;vertical-align:top;&quot; | Produce an online book chapter about a novel motivation or emotion topic.
|-
| style=&quot;vertical-align:top;&quot; | Up-to-date knowledge and skills
| style=&quot;vertical-align:top;&quot; | Utilise the most relevant psychological theory and research to address a practical question.
|-
| style=&quot;vertical-align:top;&quot; | Solve problems via thinking
| style=&quot;vertical-align:top;&quot; | Use critical thinking to explain how psychological science can address real-world problems.
|-
| rowspan=&quot;3&quot; style=&quot;vertical-align:top;&quot; | '''Be a global citizen'''
| style=&quot;vertical-align:top;&quot; | Informed and balanced
| style=&quot;vertical-align:top;&quot; | Provide a balanced, critical chapter which is accessible to a lay audience.
|-
| style=&quot;vertical-align:top;&quot; | Communicate diversely
| style=&quot;vertical-align:top;&quot; | Collaborate with peers to communicate knowledge openly with a global audience.
|-
| style=&quot;vertical-align:top;&quot; | Creative use of technology
| style=&quot;vertical-align:top;&quot; | Learn how to collaborate using wiki technology.
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Engage in new ideas
| style=&quot;vertical-align:top;&quot; | Engage in a collaborative learning culture by incorporating feedback and suggestions.
|-
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Experience project work in a collaborative, online editing environment.
|}

==Instructions==
The following instructions should be used to guide the development of the book chapter.

===Theme===
* Chapters should fit the book theme which is &quot;understanding and improving our motivational and emotional lives using psychological science&quot;

===Audience===
* The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise.

===Wikiversity===
* Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name

===Topic===
* The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]]

===Collaboration and feedback===
* Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others)
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread&lt;!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)--&gt;)
* Feedback is usually best placed on the chapter's wiki discussion page
* Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]]

===Length (word count)===
{{Anchor|Wordcount}}{{Anchor|Word count}}
* There is no minimum length
* Maximum 4,000 words
** There is no additional 10% allowance
** Words beyond the maximum will not be considered for marking purposes
** Count everything from top to bottom of the editable page (in view mode, not edit mode):
*** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links
** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor
* If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]]

===Submission===
* Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]
Book chapters will be marked against the following criteria.

===Overview (5%)===
* Scenario: Provide an engaging scenario or case study in a feature box, with an illustrative figure
* Problem statement: Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved
* Focus questions: Establish [[/Focus questions|focus questions]] which align with the sub-title and subsequent heading structure

===Theory (20%)===
* Clearly explain the theoretical framework for understanding the topic
* Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories.
* Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Clearly explain and apply the theory(ies)
* Include illustrative examples, such as case studies
* Demonstrate a critical perspective

===Research (25%)===
* Clearly explain how key, peer-reviewed research findings apply to the problem
* Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals)
* Include relevant major reviews (such as systematic reviews and meta-analyses)
* [[w:Critical thinking|Critical analyse]] the key research findings, including limitations and implications

===Integration (10%)===
* Integrate discussion of theory and review of relevant research
* Use research to critically inform interpretation and application of the theory(ies)

===Conclusion (5%)===
* Clear and concise communication of key points and take-home messages
* Aligned with the subtitle and focus questions, with implications for the [[Motivation and emotion/Book/Theme|book theme]]

===Style (20%)===
* Overall
** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]]
** [[/Readability|Readable]] for a layperson interested in psychological science
** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information
** Address an international audience (i.e., avoid an overly local or national perspective)
** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles
** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment)
** Correct grammar (e.g., see [[/Writing tips|writing tips]])
* Structure
** Use a logical heading structure that aligns with the focus questions
** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings
** Use the default heading style (e.g., do not add italics and/or bold)
** Sub-headings are optional
*** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings.
*** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections.
* Sentences
** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., &quot;it&quot;, &quot;they&quot;). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., &quot;I&quot; and &quot;we&quot;) and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., &quot;you&quot;) can work well.
* Paragraphs
** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs.
** Paragraphs flow logically
* Use APA style (as much as reasonably possible), paying particular attention to:
** citations
** references (especially capitalisation, italicisation, and providing hyperlinked dois)
** table and figure captions
** quotes (include page numbers)
* Citations
** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice.
** Maximum of 3 citations per point (i.e., avoid 4 or more citations together).
* References
** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style.
** Non-academic sources are not used in references. They can be included in the external links section.

===Learning features (5%)===
* Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the &quot;See also&quot; and &quot;External links&quot; sections
* Case studies
** Include 1 or more examples, scenarios, or case studies
** They can be true (if so, include citations) or fictional
** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages
** Present in a feature box and include a figure
** Consider using a &quot;progressive case study&quot; (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes).
** Examples of chapters which make effective use of case studies:
*** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019)
*** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019)
*** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019)
*** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019)
* [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]]
** Use to highlight key information, but avoid overuse
** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option.
* [[Motivation and emotion/Wikiversity/Figures|Figures]]
** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts
** Figures are accompanied by explanatory captions and be cited at least once in the main text
** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]).
* [[Help:Links|Links]]
** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example:
*** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is &lt;nowiki&gt;[[w:Emotion|emotion]]&lt;/nowiki&gt;). It is also possible to link to a section on this same page e.g., &amp;lt;nowiki&amp;gt;[[#Overview|Overview]]&amp;lt;nowiki&amp;gt; will link to the Overview section.
*** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages.
** See also
*** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles
*** Include source in parentheses
** External links
*** Provide at least three links to high quality, relevant external resources
*** Include author and/or source in parentheses
** Published academic sources belong in References
* [[Motivation and emotion/Wikiversity/Tables|Tables]]
** Use accompanying tables to help organise information and communicate concepts to readers
** Tables are accompanied by explanatory APA style captions and are cited in the body text
* [[Help:Quiz|Quizzes]]
** Quiz questions or reflection questions encourage reader engagement
** Focus on core concepts (esp. take-home messages) rather than trivia
** Consider incorporating throughout the chapter
{{anchor|Socialcontribution}}

{{anchor|Social contribution}}
===Social contribution (10%)===
* '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include:
** '''Edits''': Direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]].
** '''Comments''': Feedback provided on book chapter [[Help:Talk page|talk page]]s
** '''Media uploads''': Upload free-to-use educational images to [[commons:|Wikimedia Commons]]
** '''{{Motivation and emotion/Canvas}} discussion posts'''
* '''Evidence''':
** Provide a [[mw:Help:Lists|numbered list]] with summaries of contributions on your [[Help:User page|Wikiversity user page]], and '''direct links''' that show each change or contribution.&lt;!-- To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Summarise each edit on your user page (in a section called &quot;Social contributions&quot;) --&gt;
** More info: [[/Summarising social contributions|summarising social contributions]]
* '''Marking'''
** Marking of social contributions will be based on a combination of:
*** '''Quantity''':
**** Breadth: number of chapters contributed to
**** Channels: range of communication channels used
*** '''Quality''':
**** Depth/Extent/Thoroughness
**** Insightfulness
**** Practical value
*** '''Timeliness''' — there is generally:
**** Greater value in earlier contributions
**** Lesser value in last-minute contributions
** Marks will be allocated to each clearly evidenced contribution as follows:
*** Minor &lt;= 0.25
*** Moderate 0.50
*** Major 1.00
*** Very significant &gt; 1.00
* '''Bonus marks'''
** Up to 5 bonus marks (out of 100 for the book chapter) will be awarded in exceptional circumstances where social contributions are above and beyond those required for HD-level.
** Approximately 5% of students receive social contribution bonus marks.
** This could include extensive copyediting of multiple chapters and/or regular feedback or support on multiple chapter discussion pages. It may also involve substantial activity on the UCLearn discussion forum.

==Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1NUNjMCM5xOVsHj1LJuRnK8BgEdXPn4TbRf-Q6PH3oXg/edit?usp=sharing Marking rubric]}}

==Examples==
Examples of high quality [[Motivation and emotion/Book|motivation and emotion book chapters]]:
* [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022)
* [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016)
* [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019)
* [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019)

Note that as of 2025, chapters no longer include multimedia presentations.

For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]&lt;!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] --&gt;.

==Licensing==
Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener.

==See also==
* [[/FAQ/]]
* [[Motivation and emotion/Book|Previous chapters]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]]
** [[Template:MEBF|Feedback template]]
** [https://docs.google.com/document/d/15NXyApC5i6svEcJAnXv9QeRA-cbojy1m8lKReZgBH60/edit?usp=sharing Marking rubric]
* [[#Socialcontribution|Social contributions]]
** [[/Search for chapters to improve/]]
** [[/Summarising social contributions/]]
* [[Motivation and emotion/Tutorials|Tutorials]]
&lt;!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --&gt;
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]]
** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]]
* Wikiversity
** [[/Feature boxes/]]
** [[/Figures/]]
** [[How to find free-to-use images|Find free images]]
** [[/Tables/]]
* [[/Writing tips/]]
** [[/How to handle a lack of information/|Handling a lack of information]]
** [[/Readability|Readability]]
** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
** [[/Word count|Word count]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Chapter| ]]
[[Category:Motivation and emotion guidelines]]</text>
      <sha1>2lf2m7vp8t06qzotk49kh2scj1qnaae</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Numerical Methods</title>
    <ns>0</ns>
    <id>119778</id>
    <revision>
      <id>2819215</id>
      <parentid>2818595</parentid>
      <timestamp>2026-07-24T05:49:33Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2819215</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="ce2d9idj6z3i17pkvz25c0wm6wgviag" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260721.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>ce2d9idj6z3i17pkvz25c0wm6wgviag</sha1>
    </revision>
    <revision>
      <id>2819217</id>
      <parentid>2819215</parentid>
      <timestamp>2026-07-24T05:51:31Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2819217</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="n8k2z0noq475cetgwcv3libw8fm6ftl" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260722.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>n8k2z0noq475cetgwcv3libw8fm6ftl</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2819137</id>
      <parentid>2819048</parentid>
      <timestamp>2026-07-23T13:37:06Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2819137</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3176" sha1="111e3b9djyqnr98x42wiubwg9albdfp" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260723.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20260723.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>111e3b9djyqnr98x42wiubwg9albdfp</sha1>
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  </page>
  <page>
    <title>Python programming in plain view</title>
    <ns>0</ns>
    <id>212733</id>
    <revision>
      <id>2819159</id>
      <parentid>2819032</parentid>
      <timestamp>2026-07-23T16:37:51Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2819159</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2696" sha1="1hjmmr1zq769znc2mm2vwkm2xpkc1ab" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260723.pdf |A]], [[Media:Python.Work2.Library.1B.20260720.pdf |B]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
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  <page>
    <title>Algebra 1/Unit 1: Introduction To Algebra</title>
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      <contributor>
        <username>Evan Mercer</username>
        <id>3071189</id>
      </contributor>
      <comment>Namjoel’s edit was reverted because it introduced unclear and unprofessional writing, poor spelling, and bad grammar, repeated the jellybean example several times, added confusing metaphors like “plus is married to minus,” included mathematically incorrect lines such as “10+x = 10+10 = 10,” used run‑on sentences, mixed unrelated storytelling about apples and houses, and replaced clear instructional steps with disorganized, repetitive content that reduced the lesson’s educational quality.</comment>
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      <text bytes="22561" sha1="gknz0q5kqddt3k7hh2h8972g41l5nnb" xml:space="preserve">[[File:Quadratic formula.svg|thumb|right|An example of a Algebra formula (quadratic formula)]]
{{mathematics}}
{{secondary education}}
{{lesson}}
{{complete}}

'''Algebra''' (from the Arabic word &quot;al-jabr&quot; (الجبر), meaning &quot;reunion of broken parts&quot;) can feel like quite a complicated language of mathematics. However, as time goes on, completing Algebra will get easier and easier until it's a breeze. Completing Algebra takes true dedication with a worthwhile reward. This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is &lt;small&gt;IMPORTANT&lt;/small&gt; that you review through these warm ups and get comfortable in solving them to lay a strong foundation for understanding larger topics later on.

Without further do, let's dig right into this!
==Algebra==
===What is Algebra?===
[[File:AlgebraJournalWork11-14-16.jpg|thumb|left|You might have to do this much work for a small answer!]]In Algebra, we use letters to represent number or a amount of something that is not known yet. This called a '''pronumeral''' or a '''variable.''' 

Imagine you have a bag full of jellybeans on a table; with 10 green jelly beans and a unknown amount of blue jellybeans. Let's call the blue jellybeans x. Well done, this is a pronumeral they are that simple.  Now, your friend comes over, tells you there is 20 total jellybeans in the bag.  

How many blue jellybeans are there?  

The core concept of algebra is the equal sign (=). Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced.   

To find the unknown number, you need to get the letter completely by itself. You do this by using inverse operations (doing the opposite).

* Addition (+) and Subtraction (-) are opposites.
* Multiplication (times*, ×) and Division÷, /) are opposites.

To work out the number of blue jellybeans follow these steps: 

x+10=20

# Identify the goal: We want x by itself.
# See the obstacle: There is a +10 next to the x.
# Now take away ten and add the opposite to the other side of the equation. 

x=20-10

4. Do 20-10

x=10       Good job.

Checking Your Work

In maths, you should always check your work. You can do this by working if the original equation equals the same number now that you know the pronumeral or variable. 

Does (10 + x (10) = 20)? Yes! Your answer is correct.  

'''Example two'''

In algebra, a fraction line means division. So, this equation means &quot;x (unknown number) divided by 4 equals 3.&quot;

&lt;math&gt;\tfrac{x}{4}&lt;/math&gt;  = 3
# Identify the goal: Get x by itself.
# See the obstacle: The x is being divided by 4.
# Do the opposite: The opposite of division is multiplication. Multiply both sides by 4, because whatever you do to one side you must do to the other to balance the equation 

x = 3*4

x=12

Check Your Work

Put 12 back into the original equation:

* Does 12 / 4 = 3?   Yes! The answer is correct.

===== '''Important notes:''' =====
* Can be called variable OR pronumeral.
* And can be any letter from a to z.
* Solving for (pronumeral here) e.g., &quot;solving for x&quot; means finding the pronumeral.

====== '''Fun Fact''' ======
The letter that is most commonly used for variables is x and the reason for this dates back to the origin of Algebra itself; Muhammad ibn Musa al-Khwarizmi, often called one of the main &quot;founders&quot; of Algebra you could say, used to call the unknown pronumeral &quot;'''shay'''&quot;. &quot;'''Shay&quot;''' comes from the Arabic word '''شَيْء''', which essentially means &quot;thing&quot;. When Al-Khwarizmi's works were translated to Latin in medieval Spain, &quot;shay&quot; was translated as '''&quot;xay&quot;,''' since the letter x was pronounced as &quot;sh&quot; in Spain. Later on, this word &quot;'''xay&quot;''' got abbreviated to &quot;'''x&quot;''' to represent the symbol of the unknown, so we normally use x for standard questions. For more information, visit this [https://www.pbs.org/empires/islam/innoalgebra.html PBS] page.

= Algebra problems =
Solve for x.&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{''x'' − 9 = 20
|type=&quot;{}&quot;}
''x''={ 29_3 }

{''x'' − 3 = 6
|type=&quot;{}&quot;}
''x''={ 9_3 }

{''x'' + 5 = 15
|type=&quot;{}&quot;}
''x''={ 10_3 }

{''x'' + 17 = 23
|type=&quot;{}&quot;}
''x''={ 6_3 }

{4''x'' = 12
|type=&quot;{}&quot;}
''x''={ 3_3 }

{''x''/2 = 0.5
|type=&quot;{}&quot;}
''x''={ 1_3 }

{''x''/50 = 2
|type=&quot;{}&quot;}
''x''={ 100_3 }

{''x''/9 = 5
|type=&quot;{}&quot;}
''x''={ 45_3 }
&lt;/quiz&gt;

Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that...

==Arithmetic==
[[File:Multiply 4 bags 3 marbles.svg|thumb|right|4 x 3 = 12 (multiplication)]]
'''Arithmetic''' has to deal with elementary/basic levels of math, such as division, multiplication, subtraction, and addition. Basically, just working with numbers.

This SHOULD be a level familiar with you. If you are not familiar with arithmetic math/rules, then PLEASE review through Arithmetic, as you won't survive even the 1st step of Algebra. Trust me, the basics are THAT important.
===Fractions===
[[File:Cake quarters.svg|thumb|left|A Cake with fractions]]
'''Fractions''' (from Latin ''fractus'', &quot;broken&quot;) are parts of a whole. On the left side in the image of the cake, there is only &lt;math&gt;3/4&lt;/math&gt;'s of the cake showing, the other &lt;math&gt;1/4&lt;/math&gt; has been eaten/taken away.

The number, ''3'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''numerator''' (Numerator: Number at the top, tells us of how much of the number is being talked about/being used). The number, ''4'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''denominator''' (Denominator: Number showing the all time total).
;
;Simplest form/reduced form
A reduced form of a fraction is a fraction that cannot be divided by any number other than 1, and the denominator is greater than 1. So &lt;math&gt;2/4&lt;/math&gt; is NOT in simplest form, since we can divide 2 and 4, by 2... which results in the following number: &lt;math&gt;1/2&lt;/math&gt;. Though, not every fraction can be divided by 2, there are fractions, such as: &lt;math&gt;5/35&lt;/math&gt;, &lt;math&gt;7/21&lt;/math&gt;, and &lt;math&gt;30/5&lt;/math&gt;. The two first fractions are not divisible by 2, and &lt;math&gt;30/5&lt;/math&gt; can not be divided by 2 on both sides, but only on &lt;math&gt;30&lt;/math&gt;. It's important to simplify as if you were in a test, your teacher will mark your problems as incorrect if you didn't simplify your fractions. Keep in mind that simplifying a fraction into its simplest/reduced form doesn't change its value, both the original (unsimplified) fraction and its reduced form represent the same exact value/quantity. So, &lt;math&gt;\tfrac{2}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{2}&lt;/math&gt; represents the same quantity, a half! 

Here, we will present a few fractions for you to simplify.
====Sample problems for ''simplifying fractions'' (use ''/'' as the fraction line)====
&lt;quiz display=simple points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{6}{8}=&lt;/math&gt;{ 3/4_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{4}{60}=&lt;/math&gt;{ 1/15_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{30}{90}=&lt;/math&gt;{ 1/3_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{8}{18}=&lt;/math&gt;{ 4/9_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{9}{72}=&lt;/math&gt;{ 1/8_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{64}{46}=&lt;/math&gt;{ 32/23|1 9/23_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{206}{340}=&lt;/math&gt;{ 103/170_7 }
&lt;/quiz&gt;
===== Adding or Subtracting Fractions =====
[[File:Fractionsworkalgebra.PNG|thumb|right|What we just worked on, summarized]]
To simply add or subtract fractions, make sure the denominators of the fractions you are adding or subtracting are the same. If they are not, find the least common denominator (LCD). For example, if you want to add &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; and &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, you first have to multiply the 2 in &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; by '''3''', which equals '''6'''... BUT you cannot just multiply 2 only, you also have to multiply 4 by 3, since that's what you did to 2, the denominator. If you change the denominator, you have to change the numerator. ('''This step is crucial as it allows you to preserve the same value of the fraction''' but with just a different representation)

Alright, we got that out of the way, so once we have &lt;math&gt;\tfrac{12}{6}&lt;/math&gt; + &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, we can simply add. So &lt;math&gt;12&lt;/math&gt; + &lt;math&gt;4&lt;/math&gt; = &lt;math&gt;16&lt;/math&gt;, but don't add the denominators, they stay the same. So the answer is &lt;math&gt;\tfrac{16}{6}&lt;/math&gt;, and then we simplify down to &lt;math&gt;\tfrac{8}{3}&lt;/math&gt; dividing by 2 on both the numerator and denominator. 

But... did you notice something? &lt;math&gt;\tfrac{8}{3}&lt;/math&gt;? That doesn't seem right, does it? The denominator is smaller than the numerator. When you have a fraction like this, you have to convert it to a '''mixed fraction''' (skip to [[Speak_Math_Now!/Week_1:_Introduction_To_Algebra#Improper_Fraction_--.3E_Mixed_Fraction|section 2.1.1.4]]).

===== Multiplying Fractions =====
To multiply fractions, its easiest to first simplify your fraction to simplest terms. Once you have done that, you can simply multiply the numerators and the denominators. And obviously, simplify your final product, if you can. So, we have &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; and &lt;math&gt;\tfrac{2}{6}&lt;/math&gt;. You could multiply the numerators and denominators straight away and simplify at the end if you are comfortable, but to make it easier and clearer, we should simplify the fractions first. We simplify 6 and 8 by dividing both by 2, we also divide 2 and 6 by 2. So the fractions are now &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{3}&lt;/math&gt;. You simply multiply those two fractions by multiplying the numerator by the numerator, and doing the same for the denominators. After completing this process, you will get a solution (in fraction form). &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; × &lt;math&gt;\tfrac{1}{3}&lt;/math&gt; &lt;math&gt;=&lt;/math&gt; &lt;math&gt;\tfrac{3}{12}&lt;/math&gt;. 

&lt;math&gt;\tfrac{3}{12}&lt;/math&gt; is not going to be our final product, though, since we can simplify the fraction by dividing the fraction by 3, which results in &lt;math&gt;\tfrac{1}{4}&lt;/math&gt;.

===== Dividing Fractions =====
There is an interesting twist when it comes to dividing fractions. You have to turn the fraction you want to divide by (second fraction) upside-down, also known as &quot;Keep, Change, Flip&quot; where you keep the first fraction the same, change the operation to multiplication, and replace the second fractions numerator with the denominator and the denominator with the numerator. Not only that, you have to turn the division symbol (÷) into a multiplication symbol (× or •). After that, you use your skills you learned in multiplying a fraction, and you multiply both of the fractions. Simplify if you need to.

So, &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; ÷ &lt;math&gt;\tfrac{7}{12}&lt;/math&gt;. Change the division symbol to a multiplication symbol, and turn the fraction you want to divide by upside-down (the upside-down fraction is known as a '''reciprocal'''). So &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; × (or •) &lt;math&gt;\tfrac{12}{7}&lt;/math&gt;. Multiply the numerators and denominators. The answer is &lt;math&gt;\tfrac{72}{56}&lt;/math&gt;, simplified down to &lt;math&gt;\tfrac{9}{7}&lt;/math&gt;.

===== Improper Fraction --&gt; Mixed Fraction =====
Divide the numerator by the denominator. The '''quotient''' (result of the division taking place/number above the division line) will be the whole number of the mixed fraction, while the numerator will be the remainder. The denominator remains unchanged, so don't change the denominator at all!

{{notice|If you would like to take the quiz on Fractions, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Fractions Quiz]]'''}}

See also: https://www.tes.com/lessons/bJieZ4sFPJbSTw/fractions-4-mixed-numbers-and-improper-fractions

===Decimals===

Ever wondered how to write 8&lt;math&gt;\tfrac{47}{100}&lt;/math&gt; as a decimal? Well, you've got the answer: 8.47! How did we get that answer? Let's look at a few more and maybe you'll see the pattern:
# 6&lt;math&gt;\tfrac{98}{100}&lt;/math&gt; = 6.98
# 2&lt;math&gt;\tfrac{56}{100}&lt;/math&gt; = 2.56
# 9&lt;math&gt;\tfrac{27}{100}&lt;/math&gt; = 9.27
# 5&lt;math&gt;\tfrac{83}{100}&lt;/math&gt; = 5.83
You see? We simply put the mixed number in front of the dot, and with the numerator, we slap that behind the dot! Throw out the 100, it's not important when building your decimal. 

Decimals are all about place value, the value of a number in a specific place in a number. So, when we have &lt;math&gt;6.72&lt;/math&gt;, the &lt;math&gt;6&lt;/math&gt; is in the Ones place. Now, let's throw &lt;math&gt;9&lt;/math&gt; in the tens place, which is 10 times bigger than the Ones place: &lt;math&gt;96.72&lt;/math&gt;. But... that's doesn't seem enough, does it? Let's throw in a &lt;math&gt;6, 2, 8&lt;/math&gt; and a &lt;math&gt;3&lt;/math&gt; in there! And now, we have: &lt;math&gt;628,396.72&lt;/math&gt;.

Woah! That's a pretty big number, but we can easily break this number down to it's place value. Let's do it!

So, our number, &lt;math&gt;628,396.72&lt;/math&gt;, is the number we need to break down. Let's start from the decimal point, and move left:
* The number &lt;math&gt;6&lt;/math&gt; is in the Ones place. '''x10'''
* The number &lt;math&gt;9&lt;/math&gt; is in the Tens place. '''x10'''
* The number &lt;math&gt;3&lt;/math&gt; is in the Hundreds place. '''x10'''
* The number &lt;math&gt;8&lt;/math&gt; is in the Thousands place. '''x10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten thousands place. '''x10'''
* The number &lt;math&gt;6&lt;/math&gt; is in the Hundred Thousands place.

Now we have broken up the numbers left of the decimal--What about the numbers on the ''right''? Let's throw in a &lt;math&gt;5, 2, 4&lt;/math&gt; and a &lt;math&gt;7&lt;/math&gt;. Now, we have &lt;math&gt;628,396.725,247&lt;/math&gt;. Let's break this number up like we did above.

So, our number, &lt;math&gt;628,396.725,247&lt;/math&gt;, is the number we need to break down. This time, we need to start on the decimal point, and move ''right'':
* The number &lt;math&gt;7&lt;/math&gt; is in the Tenths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Hundredths place. '''x-10'''
* The number &lt;math&gt;5&lt;/math&gt; is in the Thousandths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten Thousandths place. '''x-10'''
* The number &lt;math&gt;4&lt;/math&gt; is in the Hundred Thousandths place. '''x-10'''
* The number &lt;math&gt;7&lt;/math&gt; is in the Millionths place. 

We have just now gone over the importance of Place Value in the Decimal World. Now, we will go into how to work with decimals, in the Decimal World!

See also: http://www.shmoop.com/fractions-decimals/place-value-naming-decimals.html

==== Adding/Subtracting Decimals ====
To add decimals, in addition column-style, put the decimals in its place with the decimals lined up. Then simply add on. So, for &lt;math&gt;1.5&lt;/math&gt; + &lt;math&gt;2.5&lt;/math&gt; we'd line up the decimal points. But, if we had a problem like &lt;math&gt;1.15&lt;/math&gt; + &lt;math&gt;2.0&lt;/math&gt;, we'd add a &lt;math&gt;0&lt;/math&gt; after the &lt;math&gt;0&lt;/math&gt; that is behind the decimal. Adding a zero to a place in a decimal means &quot;no value&quot;. So &lt;math&gt;10&lt;/math&gt; basically means no ones, and &lt;math&gt;100&lt;/math&gt;, means no ones or hundreds. Same things goes for subtracting as well folks.
=====Sample problems for ''adding/subtracting decimals''=====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
6.8 - 2.5 = { 4.3_6 }

{
|type=&quot;{}&quot;}
3.4 + 5.6 = { 9_6 }

{
|type=&quot;{}&quot;}
9 + 4.50 = { 13.5_6 }

{
|type=&quot;{}&quot;}
41.89 + 25.00 = { 66.89_6 }

{
|type=&quot;{}&quot;}
9.01 + 3.089 = { 12.099_6 }

{
|type=&quot;{}&quot;}
10.90 + 11.1 = { 22_6 }

{
|type=&quot;{}&quot;}
9.5 + 3.44 = { 12.94_6 }

{
|type=&quot;{}&quot; coef=&quot;2.5&quot;}
9.00 x 2.00 = { 18_6 } 
&lt;/quiz&gt;

==== Multiplying Decimals ====
[[File:9.82x5.73 multiplication image.svg|thumb|A visual representation of the multiplication example]]
Multiplying decimals isn't as hard as it really seems to be. So, we have &lt;math&gt;9.83&lt;/math&gt; × &lt;math&gt;5.73&lt;/math&gt;. For most people, column multiplication is a lot easier than side-by-side multiplication. That being mentioned, let us column these numbers:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;× &lt;math&gt;5.73&lt;/math&gt;
-------

Now that we have our problem, we should simply ignore the decimal points and just multiply as usual, so you should get this answer once you are done with that (remember to add a zero (and grow with zeros in each line) to each and every line of addition):

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt;   &lt;math&gt;68810&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt; &lt;math&gt;491500
&lt;/math&gt;
-------

With the simple usage of addition, we should get:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;   &lt;math&gt;+&lt;/math&gt;  &lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+
&lt;/math&gt; &lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;563259&lt;/math&gt;

Now, we need to bring back our handy dandy decimal point, but where? In &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;, there are FOUR numbers in these 2 numbers overall that are behind the decimal point (in each number, there are two numbers behind the decimal points). So, we have &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;. Now, that totals up to four numbers overall behind the decimal point. So in &lt;math&gt;563259&lt;/math&gt;, we need to move the decimal point four times (beginning from the right). So watch as follows:

&lt;math&gt;563259.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56325.9&lt;/math&gt;&lt;br&gt;
&lt;math&gt;5632.59&lt;/math&gt;&lt;br&gt;
&lt;math&gt;563.259&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56.3259&lt;/math&gt;

That simple. Now, review your work, your whole work should look like this:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;       × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;    &lt;math&gt;+&lt;/math&gt;&lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+&lt;/math&gt;&lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;56.3259&lt;/math&gt;

==== Dividing Decimals ====
;Dividing a decimal by a whole number
If you want to divide a decimal by a whole number, you should divide the 2 numbers, omitting the decimal point. After you are done dividing, add the decimal point to the '''quotient''' (final product/answer at the top of the long division symbol). The decimal should be right above the decimal point in the '''dividend''' (number in the box/number that is being divided). It's quite easy and simple, as long as you know how to do long division and if you are still familiar with long division. 

Hey, this seems ''too'' easy--Let's figure out how to divide a decimal by a decimal!
;Dividing a decimal by a decimal
The trick to dividing a decimal by a decimal is to shift the decimal point as many times as it gets to a whole number, so follow along: &lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt;. Now, we simply move the decimal point as many times as we need to make the number we are going to use to divide 69.45 a whole number, so watch as followed:&lt;br&gt;
&lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;694.5&lt;/math&gt; ÷ &lt;math&gt;57.8&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;6945&lt;/math&gt;. ÷ &lt;math&gt;578&lt;/math&gt;.


Now that we have finally got our dividend a whole number (and now our first number that we are going to divide), we can go ahead and divide normally (using long division). In the end, &lt;math&gt;69.45&lt;/math&gt; divided by &lt;math&gt;5.78&lt;/math&gt; should get you &lt;math&gt;12.0155709&lt;/math&gt;!

A pretty simple one we could go is &lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;, here, we simply move our dots like so:&lt;br&gt;
&lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;04.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;4&lt;/math&gt;&lt;br&gt;
Then, we can simply divide, heck... we don't even need to do long division! The answer should pop in your head, which is &lt;math&gt;16&lt;/math&gt;.

{{notice|If you would like to take the quiz on Decimals, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Decimals Quiz]]'''}}

===Percentages===
A good definition of &quot;percent&quot; is a fraction in which the denominator is the number &lt;math&gt;100&lt;/math&gt;. For example, the numbers &lt;math&gt;59%&lt;/math&gt;, &lt;math&gt;63%&lt;/math&gt;, &lt;math&gt;91%&lt;/math&gt;, and &lt;math&gt;85%&lt;/math&gt;, are the same as just saying &lt;math&gt;\tfrac{59}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{63}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{91}{100}&lt;/math&gt;, and &lt;math&gt;\tfrac{85}{100}&lt;/math&gt;. You could also say 59 out of 100 parts, 63 out of 100 parts, 91 out of 100 parts, and 85 out of 100 parts.
====Converting Percentages====
Now that we got the basis of percentages and how they operate, we should look into changing percentages.
===== Percentage → Decimal =====
Let's look in turning a percentage into a decimal point first. It's very simple. Let's say you have &lt;math&gt;\tfrac{9}{100}&lt;/math&gt;, which, in percentage form, is &lt;math&gt;9%&lt;/math&gt;. So, we have 9%. Now, we want to change it to a decimal (I don't know, think of a reason). We simply convert the percentage symbol into a decimal point, so like this: &lt;math&gt;9.&lt;/math&gt;. Now, we have &lt;math&gt;9.&lt;/math&gt;, so then we move the decimal number two places to the left, like so: &lt;math&gt;9.&lt;/math&gt; → &lt;math&gt;.9&lt;/math&gt; → &lt;math&gt;.09&lt;/math&gt;. So now, we have &lt;math&gt;0.09&lt;/math&gt;. We added the 2 zeros in because there is no value in the tenths place, and because &lt;math&gt;.09&lt;/math&gt; does not look quite right. Looks a bit off.
===== Samples problems for ''converting percentages to decimals'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
59% = { 0.59_5 }

{
|type=&quot;{}&quot;}
63% = { 0.63_5 }

{
|type=&quot;{}&quot;}
91% = { 0.91_5 }

{
|type=&quot;{}&quot;}
85% = { 0.85_5 }

{
|type=&quot;{}&quot;}
9% = { 0.09_5 }

{
|type=&quot;{}&quot;}
9834% = { 98.34_5 }

{
|type=&quot;{}&quot;}
20% = { 0.2_5 }

{
|type=&quot;{}&quot;}
4% = { 0.04_5 }

{
|type=&quot;{}&quot;}
7.6% = { 0.076_5 }

{
|type=&quot;{}&quot;}
6% = { 0.06_5 }
&lt;/quiz&gt;
===== Decimal → Percentage =====
Now to convert a decimal into percentage we essentially do the complete opposite. We have &lt;math&gt;98.34&lt;/math&gt;. We need this to be a percentage (easier to read). Move the decimal point two places to the right. So, watch: &lt;math&gt;98.34&lt;/math&gt; → &lt;math&gt;983.4&lt;/math&gt; → &lt;math&gt;9834.&lt;/math&gt; --Now, we have &lt;math&gt;9834.&lt;/math&gt;, but the decimal point, since it's now a percentage, should not be there, but instead, a percentage should talk the decimal point's place. Now, we have our final result of &lt;math&gt;9834%&lt;/math&gt;.

==== Finding percent of a number ====
[[File:Universität Bonn.jpg|thumb|right|Would this be the fictional university these students were trying to get accepted to?]]
So, 95 students applied to a university (the fictional [[User:Atcovi/Mustafa Einhoonansebadoi University|Mustafa Einhoonansebadoi University]], for example), and only 20% of the students made it. 20%? What? With this in mind, we want to find &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt;. We take the percentage, &lt;math&gt;20%&lt;/math&gt;, and divide it by &lt;math&gt;100&lt;/math&gt;. So we get &lt;math&gt;20/100&lt;/math&gt; = &lt;math&gt;.2&lt;/math&gt;. Then, we multiply &lt;math&gt;.2&lt;/math&gt; by &lt;math&gt;95&lt;/math&gt;, in which we get &lt;math&gt;19&lt;/math&gt;. So &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt; is &lt;math&gt;19&lt;/math&gt;. If that didn't make sense .2 x 95 is 2 x 95 = 190 but because we multipled 2 by ten in the tenths place to the ones place we -10 off of the number to reverse what we just did.  Therefore, only 19 students out of 95 students made it into the fictional Mustafa Einhoonansebadoi University.    

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[[Category:Speak Math Now!]]</text>
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      <contributor>
        <username>Evan Mercer</username>
        <id>3071189</id>
      </contributor>
      <comment>Moved Arithmetic to https://en.wikiversity.org/wiki/Algebra_1/Arithmetic during upgrade (temporary)</comment>
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      <text bytes="4692" sha1="4qmtutb2tc9vysjx3694l08h41o91xi" xml:space="preserve">[[File:Quadratic formula.svg|thumb|right|An example of a Algebra formula (quadratic formula)]]
{{mathematics}}
{{secondary education}}
{{lesson}}
{{complete}}

'''Algebra''' (from the Arabic word &quot;al-jabr&quot; (الجبر), meaning &quot;reunion of broken parts&quot;) can feel like quite a complicated language of mathematics. However, as time goes on, completing Algebra will get easier and easier until it's a breeze. Completing Algebra takes true dedication with a worthwhile reward. This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is &lt;small&gt;IMPORTANT&lt;/small&gt; that you review through these warm ups and get comfortable in solving them to lay a strong foundation for understanding larger topics later on.

Without further do, let's dig right into this!
==Algebra==
===What is Algebra?===
[[File:AlgebraJournalWork11-14-16.jpg|thumb|left|You might have to do this much work for a small answer!]]In Algebra, we use letters to represent number or a amount of something that is not known yet. This called a '''pronumeral''' or a '''variable.''' 

Imagine you have a bag full of jellybeans on a table; with 10 green jelly beans and a unknown amount of blue jellybeans. Let's call the blue jellybeans x. Well done, this is a pronumeral they are that simple.  Now, your friend comes over, tells you there is 20 total jellybeans in the bag.  

How many blue jellybeans are there?  

The core concept of algebra is the equal sign (=). Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced.   

To find the unknown number, you need to get the letter completely by itself. You do this by using inverse operations (doing the opposite).

* Addition (+) and Subtraction (-) are opposites.
* Multiplication (times*, ×) and Division÷, /) are opposites.

To work out the number of blue jellybeans follow these steps: 

x+10=20

# Identify the goal: We want x by itself.
# See the obstacle: There is a +10 next to the x.
# Now take away ten and add the opposite to the other side of the equation. 

x=20-10

4. Do 20-10

x=10       Good job.

Checking Your Work

In maths, you should always check your work. You can do this by working if the original equation equals the same number now that you know the pronumeral or variable. 

Does (10 + x (10) = 20)? Yes! Your answer is correct.  

'''Example two'''

In algebra, a fraction line means division. So, this equation means &quot;x (unknown number) divided by 4 equals 3.&quot;

&lt;math&gt;\tfrac{x}{4}&lt;/math&gt;  = 3
# Identify the goal: Get x by itself.
# See the obstacle: The x is being divided by 4.
# Do the opposite: The opposite of division is multiplication. Multiply both sides by 4, because whatever you do to one side you must do to the other to balance the equation 

x = 3*4

x=12

Check Your Work

Put 12 back into the original equation:

* Does 12 / 4 = 3?   Yes! The answer is correct.

===== '''Important notes:''' =====
* Can be called variable OR pronumeral.
* And can be any letter from a to z.
* Solving for (pronumeral here) e.g., &quot;solving for x&quot; means finding the pronumeral.

====== '''Fun Fact''' ======
The letter that is most commonly used for variables is x and the reason for this dates back to the origin of Algebra itself; Muhammad ibn Musa al-Khwarizmi, often called one of the main &quot;founders&quot; of Algebra you could say, used to call the unknown pronumeral &quot;'''shay'''&quot;. &quot;'''Shay&quot;''' comes from the Arabic word '''شَيْء''', which essentially means &quot;thing&quot;. When Al-Khwarizmi's works were translated to Latin in medieval Spain, &quot;shay&quot; was translated as '''&quot;xay&quot;,''' since the letter x was pronounced as &quot;sh&quot; in Spain. Later on, this word &quot;'''xay&quot;''' got abbreviated to &quot;'''x&quot;''' to represent the symbol of the unknown, so we normally use x for standard questions. For more information, visit this [https://www.pbs.org/empires/islam/innoalgebra.html PBS] page.

= Algebra problems =
Solve for x.&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{''x'' − 9 = 20
|type=&quot;{}&quot;}
''x''={ 29_3 }

{''x'' − 3 = 6
|type=&quot;{}&quot;}
''x''={ 9_3 }

{''x'' + 5 = 15
|type=&quot;{}&quot;}
''x''={ 10_3 }

{''x'' + 17 = 23
|type=&quot;{}&quot;}
''x''={ 6_3 }

{4''x'' = 12
|type=&quot;{}&quot;}
''x''={ 3_3 }

{''x''/2 = 0.5
|type=&quot;{}&quot;}
''x''={ 1_3 }

{''x''/50 = 2
|type=&quot;{}&quot;}
''x''={ 100_3 }

{''x''/9 = 5
|type=&quot;{}&quot;}
''x''={ 45_3 }
&lt;/quiz&gt;

Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that...
Arithmetic was moved to [https://en.wikiversity.org/w/index.php?title=Algebra_1/Arithmetic|Arithmetic]
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  <page>
    <title>Motivation and emotion/Assessment/Topic</title>
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      <timestamp>2026-07-23T21:23:43Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Marking and feedback */ Update for 2026</comment>
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      <text bytes="21734" sha1="pr7z4i78ep5jda2h2zh2kotrn6es6j4" xml:space="preserve">{{title|Topic development — Guidelines}}
&lt;div style=&quot;text-align: center;&quot;&gt;''Develop a chapter plan and user page''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
{{countdown
 |year       = 2025
 |month      = 08
 |day        = 14
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}}
--&gt; &lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester --&gt;{{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt;
{{TOCright}}
==Overview==
* Weight: 10%
* Due {{/Due}}
* Tasks: 
** Create a Wikiversity user account
** Select or negotiate an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]]
** Build wiki editing skills
** Develop a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] on Wikiversity which consists of:
*** Title and sub-title
*** Headings (and possibly sub-headings)
*** Key points for each section (and sub-section)
*** Figure (at least 1)
*** Learning feature (plan at least 1)
*** References (6+ relevant, high quality soruces)
*** Resources (2+ see also and 2+ external links)
** Create a Wikiversity user page
*** Introduce yourself
*** Summarise at least three different types of social contributions on your Wikiversity user page
* Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]]
* Guidance for this assignment is provided in Module 1:
** [[Motivation and emotion/Lectures/Introduction|Lecture 01]]
** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]]
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing marking rubric]
*Marks will be provided via {{Motivation and emotion/Canvas}} by Census Date (end of Week 4)
* Written feedback to help guide [[Motivation and emotion/Assessment/Chapter|book chapter]] drafting will be provided via the topic's Wikiversity discussion page 
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions

==Extensions and late submissions==
* Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence
* Submissions will be accepted up to 3 days late (-10% per day late)
* If you don't submit this assessment, withdrawal from the unit by Census Date (end of Week 4) is recommended

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|- style=&quot;vertical-align:top;&quot;
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|- style=&quot;vertical-align:top;&quot;
| Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic.
|- style=&quot;vertical-align:top;&quot;
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate your ideas by sharing a chapter plan; provide feedback on other plans.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic and submitting an on-time chapter plan.
|-
| style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Learn how to edit in a collaborative, online environment.
|}

==Instructions==
Follow these instructions for the topic development:
* Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of:
*# Title and sub-title (pre-approved or negotiated)
*# Overview
*# 3-5 other top-level headings
*#* Key points for each heading/sub-heading with citations
*#* 1+ relevant figure(s)
*#* 1+ actual or planned learning feature
*# Conclusion
*# See also
*#* 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article)
*# References (at least 6, which are cited)
*# External links
*#* 2+ external links (to external resources)
*# Wikiversity user page
*#* Self-introduction
*#* A link to the chapter being worked on
*#* Social contributions in a numbered list with a summary and direct link to evidence:
*#** 1 direct edit to improve another book chapter (past or present)
*#** 1 talk page comment on another book chapter (past or present)
*#** 1 {{Motivation and emotion/Canvas}} discussion post
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* &lt;span id=&quot;Word count&quot;&gt;Length (Word count):&lt;/span&gt; There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700).
* Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments

==Template==
{{:Motivation and emotion/Assessment/Topic/Quickstarttip}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]

Topic developments will be marked against the following criteria.

{{anchor|Title}}
===Title and sub-title (10%)===
* Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}})
* Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]]
* Do not include user name; authorship is as per the page's editing history

{{anchor|Headings}}

===Headings (10%)===
* Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links)
* Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading.
* The top-level headings should align with the sub-title and focus questions
* Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]])

{{anchor|Overview}}

===Overview (10%)===
* A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
* At least 3 bullet points outlining the &quot;problem&quot; (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]]
* 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]

{{anchor|Key points}}

===Key points (10%)===
* At least 3 bullet points per section (i.e., per heading or sub-heading)
* Overview the most relevant theory(ies), including key citations
* Overview the most relevant research, including key citations
* Provide at least 1 introductory bullet point before branching into sub-sections
* Address the problem (i.e., answer the question in the sub-title)

{{Anchor|Figure}}
===Figure (10%)===
* Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]].
* Number each figure sequentially (e.g., Figure 1, Figure 2 etc.)
* Include a descriptive caption that connects the figure to the text
* Cite each figure at least once in the main text (e.g., see Figure 1)
* Optimise image display size to make it easy to read (i.e., not too big or too small)

{{Anchor|Learning feature}}
===Learning feature (10%)===
* In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,:
** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
** Internal (wiki) links:
*** At least 1 embedded link to a relevant book chapter
*** At least 1 embedded link to a relevant Wikipedia article
* Quiz question with correct and incorrect answers
** Table with an APA style caption

{{anchor|References}}
===References (10%)===
* Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]])
* Each source should be cited at least once in the key points
* Include a balance of key theoretical and key research articles

{{anchor|Resources}}

===Resources (10%)===
* '''See also''' (Level 2 heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article)
** Provide at least 1 bullet-pointed:
*** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter
*** internal wiki link to a relevant Wikipedia page
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., Book chapter, 2023)
** Use alphabetical order
* '''External links''' (Level 2 heading): Provide at least 2 external links to key internet resources
** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles)
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., The Conversation)
** Use alphabetical order

{{anchor|User page}}

===User page (10%)===
* Create a Wikiversity user page for your user account
* Edit the user page to provide information about yourself
* Recommended headings:
** About me
** Book chapter I'm working on
*** Include an internal (wiki) link to the chapter page
** Social contributions
* Consider linking to your other online profiles

{{anchor|Social contribution}}
{{anchor|Socialcontribution}}

===Social contribution (10%)===
* On your Wikiversity user page, summarise and link to ''direct evidence'' that you made at least 3 different types of contributions:
*# direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics)
*# provided feedback by commenting on a book chapter's talk page (current or previous topics)
*# posted about motivation or emotion or the assessment tasks to the {{Motivation and emotion/Canvas}} discussion forum&lt;!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --&gt;
* [[Motivation and emotion/Wikiversity/Social contributions|More info]]

== Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]}}

==Examples==
;About
* Below are some examples of topic development submissions which received 100%
* The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions
* It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter)

;2025
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&amp;oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&amp;oldid=2726043 Elina.jean.r]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&amp;oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&amp;oldid=2727975 U3254978]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&amp;oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&amp;oldid=2739659 U3237996]

;2024
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&amp;oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&amp;oldid=2644098 U3216883]

;2023
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&amp;oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&amp;oldid=2550752 U3227684]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&amp;oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&amp;oldid=2552580 U3223114]
&lt;!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides.

;2022
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&amp;oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&amp;oldid=2420008 U3203545]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&amp;oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&amp;oldid=2419862 U3210431]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&amp;oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&amp;oldid=2420416 U3216256]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&amp;oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&amp;oldid=2419390 Icantchooseone]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&amp;oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&amp;oldid=2420715 U3217287]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&amp;oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&amp;oldid=2418874 Molzaroid]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&amp;oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&amp;oldid=2420232 Ana028]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&amp;oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&amp;oldid=2418246 U3211603]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&amp;oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&amp;oldid=2417630 U3223109]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&amp;oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&amp;oldid=2420956 U3213441]

;2021
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&amp;oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&amp;action=history U3186377]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&amp;oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&amp;action=history U3202904]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&amp;oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&amp;oldid=2313868 U3194166]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&amp;oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&amp;oldid=2313844 U3187741]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&amp;oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&amp;oldid=2310813 U3187874]
* [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&amp;oldid=2312677 U3202854]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&amp;oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&amp;oldid=2311466 U3187178]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&amp;oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&amp;oldid=2311388 J.Payten]

;2020
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&amp;oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&amp;oldid=2191350 U3170940]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&amp;oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&amp;oldid=2198005 U3190210]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&amp;oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&amp;oldid=2198079 U3160224]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&amp;oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&amp;oldid=2194797 U3190467]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&amp;oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&amp;oldid=2199869 NUMBLA0371]
;2019
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&amp;oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&amp;oldid=2051716 U3172958]
;2018
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&amp;oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&amp;oldid=1915671 U3160212]
;2017
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&amp;oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&amp;oldid=1730836 U3122707]
--&gt;

==Licensing==
Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener.

==See also==
* Structure
** [[Template:Motivation and emotion/Book chapter structure|Book chapter structure template]]
** [[/Checklist|Topic development — Checklist]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]]
** [[Template:METF|Feedback template]]
** [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]
* Tutorials
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Topic| ]]
[[Category:Motivation and emotion guidelines]]</text>
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      <text bytes="21737" sha1="6gw0bha5u5p692be86wvapsfd0n5tuf" xml:space="preserve">{{title|Topic development — Guidelines}}
&lt;div style=&quot;text-align: center;&quot;&gt;''Develop a chapter plan and user page''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
{{countdown
 |year       = 2025
 |month      = 08
 |day        = 14
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}}
--&gt; &lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester --&gt;{{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt;
{{TOCright}}
==Overview==
* Weight: 10%
* Due {{/Due}}
* Tasks
** Create a Wikiversity user account.
** Select or negotiate an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]].
** Build wiki editing skills.
** Develop a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] on Wikiversity which consists of:
*** Title and sub-title
*** Headings (and possibly sub-headings)
*** Key points for each section (and sub-section)
*** Figure (at least 1)
*** Learning feature (plan at least 1)
*** References (6+ relevant, high quality soruces)
*** Resources (2+ see also and 2+ external links)
** Create a Wikiversity user page:
*** Introduce yourself
*** Summarise at least three different types of social contributions on your Wikiversity user page
* Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].
* Guidance for this assignment is provided in Module 1:
** [[Motivation and emotion/Lectures/Introduction|Lecture 01]]
** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]]
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing marking rubric]
*Marks will be provided via {{Motivation and emotion/Canvas}} by Census Date (end of Week 4)
* Written feedback to help guide [[Motivation and emotion/Assessment/Chapter|book chapter]] drafting will be provided via the topic's Wikiversity discussion page 
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions

==Extensions and late submissions==
* Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence
* Submissions will be accepted up to 3 days late (-10% per day late)
* If you don't submit this assessment, withdrawal from the unit by Census Date (end of Week 4) is recommended

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|- style=&quot;vertical-align:top;&quot;
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|- style=&quot;vertical-align:top;&quot;
| Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic.
|- style=&quot;vertical-align:top;&quot;
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate your ideas by sharing a chapter plan; provide feedback on other plans.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic and submitting an on-time chapter plan.
|-
| style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Learn how to edit in a collaborative, online environment.
|}

==Instructions==
Follow these instructions for the topic development:
* Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of:
*# Title and sub-title (pre-approved or negotiated)
*# Overview
*# 3-5 other top-level headings
*#* Key points for each heading/sub-heading with citations
*#* 1+ relevant figure(s)
*#* 1+ actual or planned learning feature
*# Conclusion
*# See also
*#* 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article)
*# References (at least 6, which are cited)
*# External links
*#* 2+ external links (to external resources)
*# Wikiversity user page
*#* Self-introduction
*#* A link to the chapter being worked on
*#* Social contributions in a numbered list with a summary and direct link to evidence:
*#** 1 direct edit to improve another book chapter (past or present)
*#** 1 talk page comment on another book chapter (past or present)
*#** 1 {{Motivation and emotion/Canvas}} discussion post
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* &lt;span id=&quot;Word count&quot;&gt;Length (Word count):&lt;/span&gt; There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700).
* Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments

==Template==
{{:Motivation and emotion/Assessment/Topic/Quickstarttip}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]

Topic developments will be marked against the following criteria.

{{anchor|Title}}
===Title and sub-title (10%)===
* Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}})
* Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]]
* Do not include user name; authorship is as per the page's editing history

{{anchor|Headings}}

===Headings (10%)===
* Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links)
* Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading.
* The top-level headings should align with the sub-title and focus questions
* Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]])

{{anchor|Overview}}

===Overview (10%)===
* A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
* At least 3 bullet points outlining the &quot;problem&quot; (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]]
* 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]

{{anchor|Key points}}

===Key points (10%)===
* At least 3 bullet points per section (i.e., per heading or sub-heading)
* Overview the most relevant theory(ies), including key citations
* Overview the most relevant research, including key citations
* Provide at least 1 introductory bullet point before branching into sub-sections
* Address the problem (i.e., answer the question in the sub-title)

{{Anchor|Figure}}
===Figure (10%)===
* Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]].
* Number each figure sequentially (e.g., Figure 1, Figure 2 etc.)
* Include a descriptive caption that connects the figure to the text
* Cite each figure at least once in the main text (e.g., see Figure 1)
* Optimise image display size to make it easy to read (i.e., not too big or too small)

{{Anchor|Learning feature}}
===Learning feature (10%)===
* In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,:
** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
** Internal (wiki) links:
*** At least 1 embedded link to a relevant book chapter
*** At least 1 embedded link to a relevant Wikipedia article
* Quiz question with correct and incorrect answers
** Table with an APA style caption

{{anchor|References}}
===References (10%)===
* Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]])
* Each source should be cited at least once in the key points
* Include a balance of key theoretical and key research articles

{{anchor|Resources}}

===Resources (10%)===
* '''See also''' (Level 2 heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article)
** Provide at least 1 bullet-pointed:
*** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter
*** internal wiki link to a relevant Wikipedia page
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., Book chapter, 2023)
** Use alphabetical order
* '''External links''' (Level 2 heading): Provide at least 2 external links to key internet resources
** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles)
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., The Conversation)
** Use alphabetical order

{{anchor|User page}}

===User page (10%)===
* Create a Wikiversity user page for your user account
* Edit the user page to provide information about yourself
* Recommended headings:
** About me
** Book chapter I'm working on
*** Include an internal (wiki) link to the chapter page
** Social contributions
* Consider linking to your other online profiles

{{anchor|Social contribution}}
{{anchor|Socialcontribution}}

===Social contribution (10%)===
* On your Wikiversity user page, summarise and link to ''direct evidence'' that you made at least 3 different types of contributions:
*# direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics)
*# provided feedback by commenting on a book chapter's talk page (current or previous topics)
*# posted about motivation or emotion or the assessment tasks to the {{Motivation and emotion/Canvas}} discussion forum&lt;!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --&gt;
* [[Motivation and emotion/Wikiversity/Social contributions|More info]]

== Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]}}

==Examples==
;About
* Below are some examples of topic development submissions which received 100%
* The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions
* It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter)

;2025
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&amp;oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&amp;oldid=2726043 Elina.jean.r]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&amp;oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&amp;oldid=2727975 U3254978]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&amp;oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&amp;oldid=2739659 U3237996]

;2024
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&amp;oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&amp;oldid=2644098 U3216883]

;2023
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&amp;oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&amp;oldid=2550752 U3227684]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&amp;oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&amp;oldid=2552580 U3223114]
&lt;!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides.

;2022
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&amp;oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&amp;oldid=2420008 U3203545]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&amp;oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&amp;oldid=2419862 U3210431]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&amp;oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&amp;oldid=2420416 U3216256]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&amp;oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&amp;oldid=2419390 Icantchooseone]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&amp;oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&amp;oldid=2420715 U3217287]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&amp;oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&amp;oldid=2418874 Molzaroid]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&amp;oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&amp;oldid=2420232 Ana028]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&amp;oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&amp;oldid=2418246 U3211603]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&amp;oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&amp;oldid=2417630 U3223109]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&amp;oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&amp;oldid=2420956 U3213441]

;2021
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&amp;oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&amp;action=history U3186377]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&amp;oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&amp;action=history U3202904]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&amp;oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&amp;oldid=2313868 U3194166]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&amp;oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&amp;oldid=2313844 U3187741]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&amp;oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&amp;oldid=2310813 U3187874]
* [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&amp;oldid=2312677 U3202854]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&amp;oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&amp;oldid=2311466 U3187178]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&amp;oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&amp;oldid=2311388 J.Payten]

;2020
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&amp;oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&amp;oldid=2191350 U3170940]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&amp;oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&amp;oldid=2198005 U3190210]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&amp;oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&amp;oldid=2198079 U3160224]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&amp;oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&amp;oldid=2194797 U3190467]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&amp;oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&amp;oldid=2199869 NUMBLA0371]
;2019
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&amp;oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&amp;oldid=2051716 U3172958]
;2018
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&amp;oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&amp;oldid=1915671 U3160212]
;2017
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&amp;oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&amp;oldid=1730836 U3122707]
--&gt;

==Licensing==
Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener.

==See also==
* Structure
** [[Template:Motivation and emotion/Book chapter structure|Book chapter structure template]]
** [[/Checklist|Topic development — Checklist]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]]
** [[Template:METF|Feedback template]]
** [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]
* Tutorials
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Topic| ]]
[[Category:Motivation and emotion guidelines]]</text>
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      <text bytes="21740" sha1="dwbpbkahne84mmwaot4aa444k2ocr9e" xml:space="preserve">{{title|Topic development — Guidelines}}
&lt;div style=&quot;text-align: center;&quot;&gt;''Develop a chapter plan and user page''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
{{countdown
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 |month      = 08
 |day        = 14
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}}
--&gt; &lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester --&gt;{{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt;
{{TOCright}}
==Overview==
* Weight: 10%
* Due {{/Due}}
* Tasks
** Create a Wikiversity user account.
** Select or negotiate an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]].
** Build wiki editing skills.
** Develop a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] on Wikiversity which consists of:
*** Title and sub-title
*** Headings (and possibly sub-headings)
*** Key points for each section (and sub-section)
*** Figure (at least 1)
*** Learning feature (plan at least 1)
*** References (6+ relevant, high quality soruces)
*** Resources (2+ see also and 2+ external links)
** Create a Wikiversity user page:
*** Introduce yourself
*** Summarise at least three different types of social contributions on your Wikiversity user page
* Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].
* Guidance for this assignment is provided in Module 1:
** [[Motivation and emotion/Lectures/Introduction|Lecture 01]]
** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]]
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing marking rubric].
*Marks will be provided via {{Motivation and emotion/Canvas}} by Census Date (end of Week 4).
* Written feedback to help guide [[Motivation and emotion/Assessment/Chapter|book chapter]] drafting will be provided via the topic's Wikiversity discussion page.
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions.

==Extensions and late submissions==
* Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence
* Submissions will be accepted up to 3 days late (-10% per day late)
* If you don't submit this assessment, withdrawal from the unit by Census Date (end of Week 4) is recommended

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|- style=&quot;vertical-align:top;&quot;
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|- style=&quot;vertical-align:top;&quot;
| Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic.
|- style=&quot;vertical-align:top;&quot;
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate your ideas by sharing a chapter plan; provide feedback on other plans.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic and submitting an on-time chapter plan.
|-
| style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Learn how to edit in a collaborative, online environment.
|}

==Instructions==
Follow these instructions for the topic development:
* Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of:
*# Title and sub-title (pre-approved or negotiated)
*# Overview
*# 3-5 other top-level headings
*#* Key points for each heading/sub-heading with citations
*#* 1+ relevant figure(s)
*#* 1+ actual or planned learning feature
*# Conclusion
*# See also
*#* 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article)
*# References (at least 6, which are cited)
*# External links
*#* 2+ external links (to external resources)
*# Wikiversity user page
*#* Self-introduction
*#* A link to the chapter being worked on
*#* Social contributions in a numbered list with a summary and direct link to evidence:
*#** 1 direct edit to improve another book chapter (past or present)
*#** 1 talk page comment on another book chapter (past or present)
*#** 1 {{Motivation and emotion/Canvas}} discussion post
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* &lt;span id=&quot;Word count&quot;&gt;Length (Word count):&lt;/span&gt; There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700).
* Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments

==Template==
{{:Motivation and emotion/Assessment/Topic/Quickstarttip}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]

Topic developments will be marked against the following criteria.

{{anchor|Title}}
===Title and sub-title (10%)===
* Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}})
* Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]]
* Do not include user name; authorship is as per the page's editing history

{{anchor|Headings}}

===Headings (10%)===
* Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links)
* Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading.
* The top-level headings should align with the sub-title and focus questions
* Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]])

{{anchor|Overview}}

===Overview (10%)===
* A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
* At least 3 bullet points outlining the &quot;problem&quot; (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]]
* 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]

{{anchor|Key points}}

===Key points (10%)===
* At least 3 bullet points per section (i.e., per heading or sub-heading)
* Overview the most relevant theory(ies), including key citations
* Overview the most relevant research, including key citations
* Provide at least 1 introductory bullet point before branching into sub-sections
* Address the problem (i.e., answer the question in the sub-title)

{{Anchor|Figure}}
===Figure (10%)===
* Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]].
* Number each figure sequentially (e.g., Figure 1, Figure 2 etc.)
* Include a descriptive caption that connects the figure to the text
* Cite each figure at least once in the main text (e.g., see Figure 1)
* Optimise image display size to make it easy to read (i.e., not too big or too small)

{{Anchor|Learning feature}}
===Learning feature (10%)===
* In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,:
** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
** Internal (wiki) links:
*** At least 1 embedded link to a relevant book chapter
*** At least 1 embedded link to a relevant Wikipedia article
* Quiz question with correct and incorrect answers
** Table with an APA style caption

{{anchor|References}}
===References (10%)===
* Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]])
* Each source should be cited at least once in the key points
* Include a balance of key theoretical and key research articles

{{anchor|Resources}}

===Resources (10%)===
* '''See also''' (Level 2 heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article)
** Provide at least 1 bullet-pointed:
*** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter
*** internal wiki link to a relevant Wikipedia page
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., Book chapter, 2023)
** Use alphabetical order
* '''External links''' (Level 2 heading): Provide at least 2 external links to key internet resources
** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles)
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., The Conversation)
** Use alphabetical order

{{anchor|User page}}

===User page (10%)===
* Create a Wikiversity user page for your user account
* Edit the user page to provide information about yourself
* Recommended headings:
** About me
** Book chapter I'm working on
*** Include an internal (wiki) link to the chapter page
** Social contributions
* Consider linking to your other online profiles

{{anchor|Social contribution}}
{{anchor|Socialcontribution}}

===Social contribution (10%)===
* On your Wikiversity user page, summarise and link to ''direct evidence'' that you made at least 3 different types of contributions:
*# direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics)
*# provided feedback by commenting on a book chapter's talk page (current or previous topics)
*# posted about motivation or emotion or the assessment tasks to the {{Motivation and emotion/Canvas}} discussion forum&lt;!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --&gt;
* [[Motivation and emotion/Wikiversity/Social contributions|More info]]

== Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]}}

==Examples==
;About
* Below are some examples of topic development submissions which received 100%
* The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions
* It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter)

;2025
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&amp;oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&amp;oldid=2726043 Elina.jean.r]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&amp;oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&amp;oldid=2727975 U3254978]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&amp;oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&amp;oldid=2739659 U3237996]

;2024
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&amp;oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&amp;oldid=2644098 U3216883]

;2023
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&amp;oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&amp;oldid=2550752 U3227684]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&amp;oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&amp;oldid=2552580 U3223114]
&lt;!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides.

;2022
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&amp;oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&amp;oldid=2420008 U3203545]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&amp;oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&amp;oldid=2419862 U3210431]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&amp;oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&amp;oldid=2420416 U3216256]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&amp;oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&amp;oldid=2419390 Icantchooseone]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&amp;oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&amp;oldid=2420715 U3217287]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&amp;oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&amp;oldid=2418874 Molzaroid]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&amp;oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&amp;oldid=2420232 Ana028]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&amp;oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&amp;oldid=2418246 U3211603]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&amp;oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&amp;oldid=2417630 U3223109]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&amp;oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&amp;oldid=2420956 U3213441]

;2021
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&amp;oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&amp;action=history U3186377]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&amp;oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&amp;action=history U3202904]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&amp;oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&amp;oldid=2313868 U3194166]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&amp;oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&amp;oldid=2313844 U3187741]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&amp;oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&amp;oldid=2310813 U3187874]
* [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&amp;oldid=2312677 U3202854]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&amp;oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&amp;oldid=2311466 U3187178]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&amp;oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&amp;oldid=2311388 J.Payten]

;2020
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&amp;oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&amp;oldid=2191350 U3170940]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&amp;oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&amp;oldid=2198005 U3190210]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&amp;oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&amp;oldid=2198079 U3160224]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&amp;oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&amp;oldid=2194797 U3190467]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&amp;oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&amp;oldid=2199869 NUMBLA0371]
;2019
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&amp;oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&amp;oldid=2051716 U3172958]
;2018
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&amp;oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&amp;oldid=1915671 U3160212]
;2017
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&amp;oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&amp;oldid=1730836 U3122707]
--&gt;

==Licensing==
Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener.

==See also==
* Structure
** [[Template:Motivation and emotion/Book chapter structure|Book chapter structure template]]
** [[/Checklist|Topic development — Checklist]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]]
** [[Template:METF|Feedback template]]
** [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]
* Tutorials
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Topic| ]]
[[Category:Motivation and emotion guidelines]]</text>
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    <revision>
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        <id>10242</id>
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      <minor />
      <comment>/* Extensions and late submissions */</comment>
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      <text bytes="21743" sha1="1qnv4gghzaijjuehdmgwli5q3lef7mx" xml:space="preserve">{{title|Topic development — Guidelines}}
&lt;div style=&quot;text-align: center;&quot;&gt;''Develop a chapter plan and user page''
&lt;!-- ---------------------------------- ---&gt;
&lt;!-- Count down --&gt;
&lt;!-- ---------------------------------- ---&gt;&lt;!--
{{countdown
 |year       = 2025
 |month      = 08
 |day        = 14
 |hour       = 23
 |minute     = 0
 |second     = 0
 |event = this assessment is due
}}
--&gt; &lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
&lt;!-- Show this during semester --&gt;{{:Motivation and emotion/Assessment/Chapter/Contents}}&lt;/div&gt;
{{TOCright}}
==Overview==
* Weight: 10%
* Due {{/Due}}
* Tasks
** Create a Wikiversity user account.
** Select or negotiate an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]].
** Build wiki editing skills.
** Develop a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] on Wikiversity which consists of:
*** Title and sub-title
*** Headings (and possibly sub-headings)
*** Key points for each section (and sub-section)
*** Figure (at least 1)
*** Learning feature (plan at least 1)
*** References (6+ relevant, high quality soruces)
*** Resources (2+ see also and 2+ external links)
** Create a Wikiversity user page:
*** Introduce yourself
*** Summarise at least three different types of social contributions on your Wikiversity user page
* Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]].
* Guidance for this assignment is provided in Module 1:
** [[Motivation and emotion/Lectures/Introduction|Lecture 01]]
** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]]
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]

==Marking and feedback==
*Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing marking rubric].
*Marks will be provided via {{Motivation and emotion/Canvas}} by Census Date (end of Week 4).
* Written feedback to help guide [[Motivation and emotion/Assessment/Chapter|book chapter]] drafting will be provided via the topic's Wikiversity discussion page.
*Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions.

==Extensions and late submissions==
* Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence.
* Submissions will be accepted up to 3 days late (-10% per day late).
* If you don't submit this assessment, withdrawal from the unit by Census Date (end of Week 4) is recommended.

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|- style=&quot;vertical-align:top;&quot;
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|- style=&quot;vertical-align:top;&quot;
| Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour.
| Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic.
|- style=&quot;vertical-align:top;&quot;
| Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field.
| Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. 
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:

{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
! style=&quot;width:20%;&quot; | Category
! style=&quot;width:20%;&quot; | Graduate attribute
! style=&quot;width:60%;&quot; | Assessment task
|-
| rowspan=&quot;2&quot; style=&quot;vertical-align:top;&quot; | '''Be professional'''
| style=&quot;vertical-align:top;&quot; | Communicate effectively
| style=&quot;vertical-align:top;&quot; | Communicate your ideas by sharing a chapter plan; provide feedback on other plans.
|-
| style=&quot;vertical-align:top;&quot; | Display initiative and drive, and use organisation skills to plan and manage workload
| style=&quot;vertical-align:top;&quot; | Get organised by selecting a topic and submitting an on-time chapter plan.
|-
| style=&quot;vertical-align:top;&quot; | '''Be a lifelong learner'''
| style=&quot;vertical-align:top;&quot; | Evaluate and adopt new technology
| style=&quot;vertical-align:top;&quot; | Learn how to edit in a collaborative, online environment.
|}

==Instructions==
Follow these instructions for the topic development:
* Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of:
*# Title and sub-title (pre-approved or negotiated)
*# Overview
*# 3-5 other top-level headings
*#* Key points for each heading/sub-heading with citations
*#* 1+ relevant figure(s)
*#* 1+ actual or planned learning feature
*# Conclusion
*# See also
*#* 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article)
*# References (at least 6, which are cited)
*# External links
*#* 2+ external links (to external resources)
*# Wikiversity user page
*#* Self-introduction
*#* A link to the chapter being worked on
*#* Social contributions in a numbered list with a summary and direct link to evidence:
*#** 1 direct edit to improve another book chapter (past or present)
*#** 1 talk page comment on another book chapter (past or present)
*#** 1 {{Motivation and emotion/Canvas}} discussion post
* [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement
* &lt;span id=&quot;Word count&quot;&gt;Length (Word count):&lt;/span&gt; There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700).
* Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments

==Template==
{{:Motivation and emotion/Assessment/Topic/Quickstarttip}}

==Marking criteria==
[[File:Balanced scales.svg|right|125px]]

Topic developments will be marked against the following criteria.

{{anchor|Title}}
===Title and sub-title (10%)===
* Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}})
* Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]]
* Do not include user name; authorship is as per the page's editing history

{{anchor|Headings}}

===Headings (10%)===
* Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links)
* Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading.
* The top-level headings should align with the sub-title and focus questions
* Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]])

{{anchor|Overview}}

===Overview (10%)===
* A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
* At least 3 bullet points outlining the &quot;problem&quot; (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]]
* 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]

{{anchor|Key points}}

===Key points (10%)===
* At least 3 bullet points per section (i.e., per heading or sub-heading)
* Overview the most relevant theory(ies), including key citations
* Overview the most relevant research, including key citations
* Provide at least 1 introductory bullet point before branching into sub-sections
* Address the problem (i.e., answer the question in the sub-title)

{{Anchor|Figure}}
===Figure (10%)===
* Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]].
* Number each figure sequentially (e.g., Figure 1, Figure 2 etc.)
* Include a descriptive caption that connects the figure to the text
* Cite each figure at least once in the main text (e.g., see Figure 1)
* Optimise image display size to make it easy to read (i.e., not too big or too small)

{{Anchor|Learning feature}}
===Learning feature (10%)===
* In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,:
** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]
** Internal (wiki) links:
*** At least 1 embedded link to a relevant book chapter
*** At least 1 embedded link to a relevant Wikipedia article
* Quiz question with correct and incorrect answers
** Table with an APA style caption

{{anchor|References}}
===References (10%)===
* Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]])
* Each source should be cited at least once in the key points
* Include a balance of key theoretical and key research articles

{{anchor|Resources}}

===Resources (10%)===
* '''See also''' (Level 2 heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article)
** Provide at least 1 bullet-pointed:
*** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter
*** internal wiki link to a relevant Wikipedia page
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., Book chapter, 2023)
** Use alphabetical order
* '''External links''' (Level 2 heading): Provide at least 2 external links to key internet resources
** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles)
** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]]
** Include the source in parentheses after the link (e.g., The Conversation)
** Use alphabetical order

{{anchor|User page}}

===User page (10%)===
* Create a Wikiversity user page for your user account
* Edit the user page to provide information about yourself
* Recommended headings:
** About me
** Book chapter I'm working on
*** Include an internal (wiki) link to the chapter page
** Social contributions
* Consider linking to your other online profiles

{{anchor|Social contribution}}
{{anchor|Socialcontribution}}

===Social contribution (10%)===
* On your Wikiversity user page, summarise and link to ''direct evidence'' that you made at least 3 different types of contributions:
*# direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics)
*# provided feedback by commenting on a book chapter's talk page (current or previous topics)
*# posted about motivation or emotion or the assessment tasks to the {{Motivation and emotion/Canvas}} discussion forum&lt;!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --&gt;
* [[Motivation and emotion/Wikiversity/Social contributions|More info]]

== Marking rubric==
{{Notice|1=[https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]}}

==Examples==
;About
* Below are some examples of topic development submissions which received 100%
* The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions
* It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter)

;2025
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&amp;oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&amp;oldid=2726043 Elina.jean.r]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&amp;oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&amp;oldid=2727975 U3254978]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&amp;oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&amp;oldid=2739659 U3237996]

;2024
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&amp;oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&amp;oldid=2644098 U3216883]

;2023
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&amp;oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&amp;oldid=2550752 U3227684]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&amp;oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&amp;oldid=2552580 U3223114]
&lt;!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides.

;2022
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&amp;oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&amp;oldid=2420008 U3203545]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&amp;oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&amp;oldid=2419862 U3210431]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&amp;oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&amp;oldid=2420416 U3216256]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&amp;oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&amp;oldid=2419390 Icantchooseone]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&amp;oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&amp;oldid=2420715 U3217287]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&amp;oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&amp;oldid=2418874 Molzaroid]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&amp;oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&amp;oldid=2420232 Ana028]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&amp;oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&amp;oldid=2418246 U3211603]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&amp;oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&amp;oldid=2417630 U3223109]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&amp;oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&amp;oldid=2420956 U3213441]

;2021
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&amp;oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&amp;action=history U3186377]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&amp;oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&amp;action=history U3202904]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&amp;oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&amp;oldid=2313868 U3194166]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&amp;oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&amp;oldid=2313844 U3187741]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&amp;oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&amp;oldid=2310813 U3187874]
* [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&amp;oldid=2312677 U3202854]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&amp;oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&amp;oldid=2311466 U3187178]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&amp;oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&amp;oldid=2311388 J.Payten]

;2020
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&amp;oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&amp;oldid=2191350 U3170940]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&amp;oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&amp;oldid=2198005 U3190210]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&amp;oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&amp;oldid=2198079 U3160224]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&amp;oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&amp;oldid=2194797 U3190467]
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&amp;oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&amp;oldid=2199869 NUMBLA0371]
;2019
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&amp;oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&amp;oldid=2051716 U3172958]
;2018
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&amp;oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&amp;oldid=1915671 U3160212]
;2017
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&amp;oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&amp;oldid=1730836 U3122707]
--&gt;

==Licensing==
Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener.

==See also==
* Structure
** [[Template:Motivation and emotion/Book chapter structure|Book chapter structure template]]
** [[/Checklist|Topic development — Checklist]]
* Marking and feedback
** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]]
** [[Template:METF|Feedback template]]
** [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]
* Tutorials
** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]]
** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]]
* [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]]
{{Motivation and emotion/Assessment/Navigation}}
[[Category:Motivation and emotion/Assessment/Topic| ]]
[[Category:Motivation and emotion guidelines]]</text>
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      <text bytes="3198" sha1="7lxhrtrbsgofaqxnmcsbfjy67e52rps" xml:space="preserve">'''OpenStax''' (formerly OpenStax College) is a nonprofit ed-tech initiative based at Rice University. Since 2012, OpenStax has created peer-reviewed, openly licensed textbooks, which are available as free downloadable PDFs, web versions, audiobooks&lt;ref&gt;{{Cite web|url=https://openstax.org/blog/guest-post-how-audio-technology-is-creating-more-inclusive-learning|title=OpenStax {{!}} How audio technology is creating more inclusive learning|website=openstax.org|language=en-US|access-date=2025-10-20}}&lt;/ref&gt; and for a low cost in print. All textbook content is licensed under Creative Commons Attribution Licenses; specifically, the books are available under the Creative Commons Attribution-NonCommercial-ShareAlike License v4.0, which means that instructors are free to use, adapt, and remix the content, as long as they attribute OpenStax.&lt;ref&gt;[[Wikipedia: OpenStax]]&lt;/ref&gt;

The following Wikiversity resources devoted to OpenStax textbooks.  These resources also included materials available at [https://openstax.org/ '''openstax.org''']. Although some versions found on Wikiversity are out-of-date, some might find them more convenient to access.

*[[OpenStax American Government 3e]]
*[[OpenStax American Government 4e]]
*[[OpenStax University Physics|OpenStax University Physics (click to visit)]]

*[[OpenStax College Physics|OpenStax College Physics (click to visit)]]

*[[OpenStax Astronomy|OpenStax Astronomy (click to visit)]]
*[[OpenStax Astronomy 2e|OpenStax Astronomy 2e (click to visit)]]
*[[OpenStax Anatomy and Physiology 2e|OpenStax Anatomy &amp; Physiology 2e]]
*[[OpenStax Biology 2e]]
*[[OpenStax Business Ethics]]
*[[OpenStax College Success Concise]]
*[[OpenStax Concepts of Biology]]
*[[OpenStax Introduction to Anthropology]]
*[[OpenStax Introduction to Business]]
*[[OpenStax Introduction to Business 2e]]
*[[OpenStax Introduction to Political Science]]
*[[OpenStax Introduction to Sociology 3e]]
*[[OpenStax Lifespan Development]]
*[[OpenStax Nutrition for Nurses]]
*[[OpenStax Organizational Behavior]]
*[[OpenStax Principles of Economics 3e]]
*[[OpenStax Principles of Macroeconomics 3e]]
*[[OpenStax Principles of Microeconomics 3e]]
*[[OpenStax Psychology 2e]]
*[[OpenStax US History]]
*[[OpenStax world history volume 1 to 1500|OpenStax World History, Volume 1: to 1500]]

*'''OpenStax Calculus: ''' No resources have been developed, but (out-of-date) pdf versions of the three volume textbook are posted on Wikiversity at: '''[[:File:CalculusVolume1-OP.pdf|V1]]''' | '''[[:File:CalculusVolume2-OP.pdf|V2]]''' | '''[[:File:CalculusVolume3-LR.pdf|V3]]''' 

== See Also ==
* [[Wikipedia: OpenStax]]

* [https://openstax.org/ OpenStax.org]
* [https://audileo.com/ Official OpenStax Audio Textbooks]

*[[:Category:openstax textbook]]
*[https://www.facebook.com/openstax/ OpenStax Facebook page]
* [https://www.ted.com/talks/richard_baraniuk_the_birth_of_the_open_source_learning_revolution TED Talk dated 2006-02] Founder Richard Baraniuk discussing Connexions
*[[Quizbank]]
* [https://www.youtube.com/watch?v=Xog2X2SnjvQ YouTube: Importing OpenStax content into Pressbooks]

== References ==
{{reflist}}

{{subpages/List}}

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      <text bytes="8786" sha1="ea8krf9imj7h5lk6nhl8ansyaduull3" xml:space="preserve">[[w:Kernel-based_Virtual_Machine|Kernel-based Virtual Machine]] (KVM) is a virtualization module in the Linux kernel that allows the kernel to function as a hypervisor. KVM supports dynamic memory management&lt;ref&gt;https://www.linux-kvm.org/page/FAQ#Is_dynamic_memory_management_for_guests_supported.3F&lt;/ref&gt; and [[/Live Migration/]]&lt;ref&gt;https://www.linux-kvm.org/page/Migration&lt;/ref&gt; since 2007 among other features.

KVM is included in all mayor distribution including Ubuntu and RHEL (RedHat), RedHat provides 2 version of KVM, the one included with RHEL which allows to run up to 4 VM or the one included with [[Red Hat Virtualization|RHV]] with unlimited VM.&lt;ref&gt;https://www.redhat.com/en/topics/virtualization/what-is-KVM&lt;/ref&gt; 

&lt;code&gt;systemd-cgtop&lt;/code&gt; can be used for showing control groups resource utilization.

== Basic KVM commands ==


=== Config files and directories ===
&lt;code&gt;/etc/[[kVM/libvirt|libvirt]]/[[kVM/qemu|qemu]]&lt;/code&gt;

=== Information ===
* Show information by pool: &lt;code&gt;[[/virsh pool-info/]] YOUR_POOL_NAME&lt;/code&gt;
* &lt;code&gt;[[/virsh guestinfo/]]&lt;/code&gt;

=== Operations ===
* Editing Virtual Machine configuration: &lt;code&gt;[[/virsh edit/]] $VM_ID&lt;/code&gt;
* Create VM_MACHINE: &lt;code&gt;[[/virsh define/]] [[/MY_XML_VM_DEFINITION.XML/]]&lt;/code&gt;&lt;ref&gt;https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/6/html/virtualization_administration_guide/section-libvirt-dom-xml-example&lt;/ref&gt;
* Modifying CPUs: &lt;code&gt;virsh setvcpus $VM_NAME ([--maximum] [[--config] [--live] | [--current]])&lt;/code&gt;  // Can be live or offline in the next reboot &lt;ref&gt;http://wiki.stoney-cloud.org/wiki/VM_CPU_Hotplug&lt;/ref&gt;
** To check vcpu info: &lt;code&gt;[[/virsh vcpuinfo/]] $VM_NAME&lt;/code&gt;
* Modifying Memory: &lt;code&gt;[[Virsh/virsh setmem|virsh setmem]] &lt;vm_name&gt; &lt;memsize&gt;&lt;/code&gt;
* Connection by console: &lt;code&gt;virsh console VM_NAME&lt;/code&gt;. See also: [[virt-manager]]
* Clone a virtual machine: &lt;code&gt;virt-clone&lt;/code&gt;
* Create image: &lt;code&gt;[[/qemu-img/]] create&lt;/code&gt;
* Resize image: &lt;code&gt;[[/qemu-img/]] resize&lt;/code&gt;
* Create domain/virtual machine: &lt;code&gt;virt-install --autostart --check all=on --name VM_DOMAIN_NAME --memory VM_MEMORY_SIZE --vcpus VM_VCPUs --os-variant ubuntu18.04 --cdrom VM_INSTALLATION_MEDIA --disk /path/to/VM_STORAGE_POOL_NAME/VM_DOMAIN_NAME.qcow2 --noautoconsole --wait VM_CREATION_WAIT_TIME --network VM_NETWORK_SETTINGS&lt;/code&gt;
* &lt;code&gt;[[/virsh migrate/]]&lt;/code&gt;

=== Monitoring ===
* &lt;code&gt;[[/virsh domstate/]]&lt;/code&gt;
* &lt;code&gt;[[/virsh domstats/]]&lt;/code&gt;

=== KVM Networking ===
* &lt;code&gt;virsh [[virsh/net-list|net-list]]&lt;/code&gt;
* &lt;code&gt;virsh [[virsh/net-info|net-info]] YOUR_NET_NAME&lt;/code&gt;
* &lt;code&gt;virsh net-destroy YOUR_NET_NAME&lt;/code&gt;
See also: [[MacVTap]], virbr0, ovirtmgmt, [[virtio]]&lt;ref&gt;https://wiki.libvirt.org/page/Virtio&lt;/ref&gt; and [[brctl]]. 

==== Activities ====
# Read KVM Network Performance - Best Practices and Tuning Recommendations &lt;ref&gt;https://www.ibm.com/support/knowledgecenter/en/linuxonibm/liaag/wkvm/l0wkvm00_2016.htm&lt;/ref&gt;
# Learn about [[KVM]] Host Networking Configuration Choices https://www.ibm.com/support/knowledgecenter/en/linuxonibm/liaag/wkvm/wkvm_c_net.htm

=== Storage ===
* List Storage pools:
::&lt;code&gt;[[/virsh pool-list/]]&lt;/code&gt;&lt;ref&gt;http://manpages.ubuntu.com/manpages/precise/man1/virsh.1.html#storage%20pool%20commands&lt;/ref&gt;
::&lt;code&gt;virsh pool-list --all&lt;/code&gt;
* Show information by pool: &lt;code&gt;[[/virsh pool-info/]] YOUR_POOL_NAME&lt;/code&gt;
* List Volumes from a pool: &lt;code&gt;[[/virsh vol-list/]] YOUR_POOL_NAME&lt;/code&gt;
* Create pool:(RHEL 6 documentation &lt;ref&gt;https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/6/html/virtualization_administration_guide/sect-virtualization-storage_pools-creating-local_directories-virsh&lt;/ref&gt;)
&lt;pre&gt;
    virsh pool-define-as --name YOUR_POOL_NAME --type dir --target /your/path/YOUR_POOL_NAME/ 
    virsh pool-build YOUR_POOL_NAME
    virsh pool-start YOUR_POOL_NAME
    virsh pool-autostart YOUR_POOL_NAME
&lt;/pre&gt;
See also: https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/7/html/virtualization_deployment_and_administration_guide/sect-managing_guest_virtual_machines_with_virsh-storage_pool_commands

* Create volume
::&lt;code&gt;virsh vol-create-as --format qcow2 --capacity VM_DISK_SIZE --pool VM_STORAGE_POOL_NAME --name VM_SERVER_NAME.qcow2&lt;/code&gt;

=== Snapshots &lt;ref&gt;http://manpages.ubuntu.com/manpages/xenial/man1/virsh.1.html#snapshot%20commands&lt;/ref&gt; ===
Snapshots can be offline or live and internal or external:
* RedHat recommend using external snapshots using &lt;code&gt;virsh snapshot-create-as&lt;/code&gt; command. &lt;ref&gt;https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/7/html/virtualization_deployment_and_administration_guide/sect-managing_guests_with_the_virtual_machine_manager_virt_manager-managing_snapshots&lt;/ref&gt;
* Live snapshot are not supported in [[Red Hat Enterprise Linux|RHEL]] 8&lt;ref&gt;https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/8/html-single/configuring_and_managing_virtualization/index&lt;/ref&gt;

* List snapshots:&lt;code&gt;virsh snapshot-list $DOMAIN&lt;/code&gt;&lt;ref&gt;https://www.cyberciti.biz/faq/how-to-create-create-snapshot-in-linux-kvm-vmdomain/&lt;/ref&gt;
* Create snapshot: &lt;code&gt;virsh snapshot-create-as --domain {VM-NAME} --name &quot;{SNAPSHOT-NAME}&quot;&lt;/code&gt;
* Revert snapshot: &lt;code&gt;virsh snapshot-revert --domain YOUR_DOMAIN --snapshotname YOUR_SNAPSHOT_NAME --running&lt;/code&gt;

* Snapshots with blockcommit&lt;ref&gt;https://wiki.libvirt.org/page/Live-disk-backup-with-active-blockcommit&lt;/ref&gt; and blockpull&lt;ref&gt;https://kashyapc.wordpress.com/2013/01/22/live-backup-with-external-disk-snapshots-and-libvirts-blockpull/&lt;/ref&gt;

== Hybernate VM ==
* &lt;code&gt;virsh save&lt;/code&gt;

== Performance ==
* &lt;code&gt;[[systemd-cgtop]]&lt;/code&gt; can be used for showing control groups resource utilization.

== Activities ==
=== General ===
# [[/Install KVM in an Ubuntu Server/]] following https://help.ubuntu.com/community/KVM/Installation instructions:
# Connect to KVM using [[/Virt Manager/]] (&lt;code&gt;brew install [[virt-manager]]&lt;/code&gt; &lt;ref&gt;https://help.ubuntu.com/community/KVM/VirtManager&lt;/ref&gt;
# Create a KVM virtual machine with Ansible: [[DevOps/Ansible/ovirt-RHV module]]
# Read about raw files and [[qcow2]] differences: https://www.berrange.com/posts/2017/02/10/the-surprisingly-complicated-world-of-disk-image-sizes/

=== Snapshots ===
* Read KVM [[virsh]] external vs internal snapshots: https://rentry.co/644h6
* Read RedHat recommendations: https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/7/html/virtualization_deployment_and_administration_guide/sect-Troubleshooting-Workaround_for_creating_external_snapshots_with_libvirt
* Read this guide about handling snapshots: https://www.cyberciti.biz/faq/how-to-create-create-snapshot-in-linux-kvm-vmdomain/

=== Advanced ===
* [[/Extend LVM on qcow2 images/]]: https://www.technomenace.com/2014/08/how-to-extend-lvm-on-qcow2-images/
* Live Migrate a machine between two servers&lt;ref&gt;https://www.berrange.com/posts/2016/05/12/analysis-of-techniques-for-ensuring-migration-completion-with-kvm/&lt;/ref&gt;.
* [[/Offline Migrate or move a virtual machine between two KVM servers without shared storage/]]&lt;ref&gt;https://serverfault.com/questions/434064/correct-way-to-move-kvm-vm&lt;/ref&gt;: &lt;code&gt;[[virsh/virsh dumpxml|virsh dumpxml]]&lt;/code&gt; and &lt;code&gt;[[virsh/MY XML VM DEFINITION.XML|virsh define]]&lt;/code&gt;
[[Backup]]:
* [[virsh/Create Live backups]] using [[virsh/virsh blockcommit|virsh blockcommit]]
* Review [[Bacula]] Enterprise KVM plugin: http://www.bacula.lat/kvm-plugin-bacula-enterprise-quick-guide/?lang=en
* Review [[Bacula]] solution with &lt;code&gt;bpipe&lt;/code&gt;: http://www.bacula.lat/kvm-hot-online-virtual-machines-backup-with-bacula-bpipe/?lang=en
* Review Veritas [[Netbackup]] support for [[Red Hat Virtualization|RHV]]

== See also ==
* CompTIA [[IT Fundamentals/Virtualization]]
* [[w:kernel-based_Virtual_Machine]]
* [[VirtualBox]], [[multipass]]
* [[/qemu/]], [[/libvirt/]] and [[virsh]]
* &lt;code&gt;[[virtinst]]&lt;/code&gt; package&lt;ref&gt;https://github.com/virt-manager/virt-manager/tree/master/virtinst&lt;/ref&gt;, includes following python binaries: &lt;code&gt;virt-clone&lt;/code&gt;, &lt;code&gt;virt-convert&lt;/code&gt;, &lt;code&gt;[[/virt-install/]]&lt;/code&gt;&lt;ref&gt;https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/8/html/configuring_and_managing_virtualization/getting-started-with-virtualization-in-rhel-8_configuring-and-managing-virtualization&lt;/ref&gt;
* [[Red Hat Virtualization]] ([[Red Hat Virtualization|RHV]])
* [[/virt-manager/]] (&lt;code&gt;virsh console VM_NAME&lt;/code&gt;)
* [[MacVTap]] device driver
* [[/virt-sysprep/]]
* [[DevOps/Ansible|Ansible]] [[DevOps/Ansible/ovirt-RHV module|ovirt-RHV module]] (ovirt_vm) 
https://docs.ansible.com/ansible/latest/modules/ovirt_vm_module.html#ovirt-vm-module

== References ==
{{reflist}}

{{CourseCat}}</text>
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      <text bytes="682" sha1="f872qrkqvf15i49kd0knemu6u3fcj0y" xml:space="preserve">&lt;noinclude&gt;{{title|Drop-in consultation}}&lt;/noinclude&gt;
* 3.5 hours available each teaching week (Weeks 1–8, 10–13)
* 30 mins before and after [[Motivation and emotion/Lectures|lectures]] and [[Motivation and emotion/Tutorials|tutorials]]:
** Wednesdays (Virtual Room)
*** 10:00–10:30
*** 12:30–13:30
*** 14:30–15:00
*** 17:00–17:30
*** 18:30–19:00
** Thursdays (On-campus in 1C33)
*** 12:30–13:00
*** 14:00–14:30
* Open to all students—you do not need to wait for your own scheduled lecture or tutorial class
* Feel free to pop in, hang out, and chat—no question is too silly, big, or small
&lt;noinclude&gt; [[Category:Motivation and emotion/Tutorials]] &lt;/noinclude&gt;</text>
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      <comment>/* What prevents Monism Panpsychism Pantheism Theorem from being posted on wikiphilophers? */ Reply</comment>
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== Could you do another manual revert on [[Web design]] now that it’s semi-protected? ==

See https://en.wikiversity.org/w/index.php?title=Web_design&amp;diff=next&amp;oldid=1936739 (the [[Special:Contributions/2402:8100:3009:2DEE:1:2:F528:AAC9|only contribution from that IP]]). They replaced the CSS Zen Garden screenshot – which is admittedly somewhat outdated, but still more relevant to web design than a 19th century painting.
--[[Special:Contributions/78.23.192.69|78.23.192.69]] ([[User talk:78.23.192.69|discuss]]) 21:16, 29 March 2025 (UTC)
: Reverted as you proposed ([[Special:Diff/2710640|diff]]}}. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:48, 3 April 2025 (UTC)

== What prevents Monism Panpsychism Pantheism Theorem from being posted on wikiphilophers? ==

Hello, thank you for all your replies so far.

But I have responded to everything you made challenge of. I removed the empty set, I removed nothing implies nothing... and I rebutted your claim that nothing could logically exist.

Do you have any more challenges to my proof?

If not, could you add my proof to wikiphilosophers?

https://en.wikiversity.org/wiki/User:MarsSterlingTurner/What_there_is_Consciousness

I know you're an atheist. But that should not stop me from adding a proof of this quality from wikiphilosophers.... should it?

Only Christopher Langan's CTMU and Baroch Spinoza's Ethics come close to the quality of the proof that I provided... and both are considered academic material. [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 00:24, 9 April 2025 (UTC)
: [[User:MarsSterlingTurner/What there is Consciousness]] is rank nonsense.
: The question is at what length one should spend time here and effort to articulate point-by-point that your nonsense is nonsense; I don't know.
: Your user page has some red flags as well, e.g. &quot;I invented several stars and nuclear reactors&quot; and &quot;I invented a cheap and effective form of synthetic telepathy&quot;. It could also be meaningful to block you for block evasion (if it really is block evasion, but it seems very likely).
: An example of a page that was moved to user space as not good enough: [[User:TyEvSkyo/Particle Sphere Theory]], although it is pseudophysics, not pseudophilosophy. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 03:46, 9 April 2025 (UTC)
::Do you believe it's impossible to invent a nuclear reactor or envision a giant one capable of being a light and or electron source?
::Do you believe synthetic telepathy is fake or something? Feel free to search the patent office for &quot;synthetic telepathy&quot;. My invention does work, I have already applied it. and it produced extraordinary results.
::You claimed that nothing can or did exist at some time. Which to me is rank nonsense that you have not proven. Do you think your mere claims can rebut logical tautologies? [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 23:42, 9 April 2025 (UTC)
::: For the record, I requested a block at [[Wikiversity:Request custodian action#Block of MarsSterlingTurner]], which was implemented. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:26, 10 April 2025 (UTC)

A later note: by searching for &quot;nothing has the property of nothing&quot;, I found also the following 2025 discussion apparently started by MarsSterlingTurner (the user account there has a photo of the face): 
* [https://atheistdiscussion.org/forums/showthread.php?tid=9242 what pure logical tautologies can deduce], June 2025, atheistdiscussion.org
There are 5 pages of discussion. Apparently, the discussion members do show some engagement with this brand of low-grade material; they bother to respond. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:44, 28 August 2025 (UTC)

:The only argument you held against my work was that &quot;nothing can't imply nothing&quot;; but what do you think of the following;
:I know you will grant that nothing is nothing.
:But is can refer to the law of identity and therefore the biconditional; ↔
:Therefore you must grant that {}↔{}
:Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →
:Therefore you must grant {}→{}
:In other words ({}is{})→({}↔{})→({}→{}) [[User:AssumingNOTHING|AssumingNOTHING]] ([[User talk:AssumingNOTHING|discuss]] • [[Special:Contributions/AssumingNOTHING|contribs]]) 21:53, 29 August 2025 (UTC)
:: I tried to explain the matter at [[User talk:MarsSterlingTurner#Reflexivity of implication and empty set etc.]]. Further engagement with this kind of thinking/argumentation seems unlikely to be productive/helpful. I opened [[Wikiversity:Request custodian action#Block of AssumingNOTHING]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:18, 30 August 2025 (UTC)
:::This is a different argument; Did you grant the biconditional? [[Special:Contributions/2601:647:6512:3F4A:DD9D:BE18:F1E1:6F3A|2601:647:6512:3F4A:DD9D:BE18:F1E1:6F3A]] ([[User talk:2601:647:6512:3F4A:DD9D:BE18:F1E1:6F3A|discuss]]) 16:58, 30 August 2025 (UTC)
:::: No, I do not grant (or agree with) the biconditional &quot;{}↔{}&quot;, meaning &quot;the empty set if and only if empty set&quot;: the latter is nonsense, to me, and thus so is the former. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:27, 1 September 2025 (UTC)
:::::[[wikipedia:Law_of_identity|Law of identity - Wikipedia]]
:::::&quot;Modern logic
:::::In first-order logic, identity (or equality) is represented as a two-place predicate, or relation, =. Identity is a relation on individuals. It is not a relation between propositions, and is not concerned with the meaning of propositions, nor with equivocation. The law of identity can be expressed as , where x is a variable ranging over the domain of all individuals. In logic, there are various different ways identity can be handled. In first-order logic with identity, identity is treated as a logical constant and its axioms are part of the logic itself. Under this convention, the law of identity is a logical truth.&quot;
:::::If equality is a logical truth, then identity is a logical equality.
:::::Equals is logically equivalent to the biconditional. [[Special:Contributions/2601:647:6512:805F:98A3:A945:B2BD:C73A|2601:647:6512:805F:98A3:A945:B2BD:C73A]] ([[User talk:2601:647:6512:805F:98A3:A945:B2BD:C73A|discuss]]) 11:51, 1 September 2025 (UTC)
:::::: I don't know what the statement &quot;equality is a logical truth&quot; is supposed to mean or refer to. (Perhaps I should disengage; let us see.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:13, 1 September 2025 (UTC)
:::::::I mean &quot;law of equality&quot; is a logical truth as &quot;equality&quot; and &quot;identity&quot; are interchangeable and it says &quot;law of identity is a logical truth&quot; therefore &quot;law of equality is a logical truth&quot;, I said equality as a generalization for it's law.
:::::::Logical equality besides using = also can be expressed as the biconditional. Do you agree with that? [[Special:Contributions/2601:647:6512:805F:45BC:E93D:A300:BAE1|2601:647:6512:805F:45BC:E93D:A300:BAE1]] ([[User talk:2601:647:6512:805F:45BC:E93D:A300:BAE1|discuss]]) 19:35, 1 September 2025 (UTC)
:::::::: What reliable/serious sources use the term &quot;logical equality&quot; and what do they mean by it? Wikipedia is not a serious source, so I do not take [[Wikipedia: Logical equality]] seriously (it does not trace the term to a serious source, actually to any source). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:29, 2 September 2025 (UTC)
:::::::: As for &quot;identity is a logical equality&quot;: if ''logical equality'' is sameness of truth values (is it?), then this statement is false, since identity (the two-place relation) works not only for truth values but also for integers, cats, etc. (I am the author of [[An analysis of identity]].) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:37, 2 September 2025 (UTC)
:::::::: Consulting https://math.stackexchange.com/questions/3528250/what-is-the-difference-between-equality-and-logical-identity, this reinforces my request above for a definition/clarification of the term &quot;logical equality&quot;. I would also like to see seriously sourced definition of &quot;logical identity&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:47, 2 September 2025 (UTC)
:::::::::∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)
:::::::::∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={}) [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:23, 2 September 2025 (UTC)
::::::::::x↔y
::::::::::{}↔{} [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:25, 2 September 2025 (UTC)
:::::::::::[∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)]=(x↔y)
:::::::::::[∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={})]=({}↔{}) [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:27, 2 September 2025 (UTC)
::::::::::::[∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)]=(x↔y)
::::::::::::question; IF [∀x∀y[∀z((z∈x)↔(z∈y))] Then (x↔y) ?
::::::::::::x={}, y={}, z={}
::::::::::::therefore [(({}∈{})↔({}∈{}))↔({}={})]=({}↔{})
::::::::::::If (({}∈{})↔({}∈{}) then ({}↔{})
::::::::::::[[wikipedia:Axiom_of_extensionality#In_ZF_set_theory|Axiom of extensionality - Wikipedia]] [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:36, 2 September 2025 (UTC)
:::::::::::::Therefore (x=y)=(x↔y)
:::::::::::::Therefore ({}={})=({}↔{}) [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:39, 2 September 2025 (UTC)
: (Outdent) Anyone who has the first idea of the first-order logic (which is apparently employed here) must see that &quot;∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={})&quot; is sheer rubbish, even hilarious rubbish. The idea that you could take the meaningful ∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y) and get something meaningful from it by replacing all the variables with &quot;{}&quot; can hardly be more preposterous.
: This is perhaps not surprising given that this rubbish is comming from someone who, is his own words, has invented several stars (whatever that is supposed to mean). I am inclined to keep removing any more posts from the same source and keep on asking for an indef block for block evasion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:37, 3 September 2025 (UTC)
: Out of curiosity, I entered the first two lines from the above into Google Gemini 2.5 Flash:
::∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)
::∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={})
: I got the following response:
:: The second expression you wrote, ∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={}), is not a valid logical or mathematical statement. It appears to be a pattern based on the first expression, but the use of empty curly braces {} makes it meaningless. The first expression, however, is a fundamental axiom in set theory. It is a formal way of stating the Axiom of Extensionality. [...]
: Not too shabby. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:26, 3 September 2025 (UTC)
::I notice you deleted the post that corrected the issue you are claiming.
::Here is what I put in the same AI; [https://aistudio.google.com/prompts/new_chat?model=gemini-2.5-flash-image-preview Google AI Studio]
::∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)
::x={}, y={}, z={}
::therefore [({}∈{})↔({}∈{})]↔({}={})
::If (({}∈{})↔({}∈{}) then ({}↔{})
::therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]
::therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})
::therefore [({}={})→({}↔{})]
::Here was the response;
::5. [({}={})→({}↔{})]
::({}={}) is True.
::If ({}↔{}) is interpreted as True, then True → True is True.
::Summary:
::Your logical deductions seem to follow correctly, given the initial interpretation of {}∈{} as false and {}↔{} as true (representing self-equivalence or equality).
::Notice the AI accepts ({}↔{}) as true, due to my proof. [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 21:22, 3 September 2025 (UTC)
::: If you had the first idea of what you are doing, you would not need to correct your obvious nonsense (&quot;∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={})&quot;.
::: Let's play this game, though; perhaps it will be educational a bit despite the odds. I put your text above before &quot;Summary:&quot; to Gemini 2.5 Flash. The response I got is this: &quot;Your logical derivation contains several misunderstandings of both set theory and logic. Let's break down where the reasoning goes wrong. [...]&quot;.
::: However, I put &quot;Is the following accurate? ({}={})→({}↔{})&quot; to Gemini 2.5 Flash. As part of the response, it says 'The notation ({}={}) and ({}↔{}) does not have a defined meaning in formal logic.' I will note that '{}={}' does have a well-defined meaning in the first-order logic applied to the language of set theory, but &quot;{}↔{}&quot; does not. As a word of warning, Gemini gives slightly different answers when I repeat the query in new sessions; in this sense, it is not deterministic. As a further word of warning, this is an inquiry into the behavior of Gemini more than into the meaning of &quot;({}={})→({}↔{})&quot; since GenAI cannot generally be relied on/trusted.
::: Another interesting prompt is &quot;What does the following mean? ({}={})→({}↔{})&quot;. As part of a longer response, I get his:
::: 'This means:
:::: '&quot;If x is equal to y, then x has the same truth value as y.&quot;
::: 'This statement is always true in logic. If two variables or propositions are literally equal to each other, they must necessarily have the same properties, including their truth value.'
::: I don't agree with Gemini on this since x and y are not guaranteed to be truth values or objects carrying truth values. But things would improve thus: 'If x is equal to y, then x has the same truth value as y, as long as both x and y do have truth value.' However, this corrected sentence does not need to be true in Python or C++ since: &quot;equal&quot; would be interpreted as &quot;==&quot; rather than Python's is, and a class can redefine &quot;==&quot; to meaning something arbitrary and non-sensical as well as the bool casting operator, and then, in Python, possibly x == y while bool(x) != bool(y). If we try to carry this over to natural language, we may ask whether there is a significant difference between equality and identity; there is one in Python.
::: We see that Gemini contradicts itself: in response to the &quot;Is the following accurate? [...]&quot; prompt, it denies the expression has meaning, whereas in response to the &quot;What does the following mean? [...]&quot; prompt, it finds a meaning and explains it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:53, 4 September 2025 (UTC)
::::I put &quot;is the following true; ({}↔{})?&quot; in the AI
::::it said;
::::Yes, ({}↔{}) is true.
::::In propositional logic, &quot;p ↔ q&quot; (p if and only if q) is true if and only if p and q have the same truth value.
::::In this case, both sides of the biconditional are &quot;{}&quot;, which represents an empty set. When considering the truth value of an empty set, it is often interpreted as being vacuously true or having a consistent truth value in certain contexts (like in set theory or type theory, an empty type might be considered true if it implies nothing).
::::However, in the simplest interpretation for a basic logical statement like this, if we treat &quot;{}&quot; as a proposition, and both sides are identical, then they must have the same truth value. Therefore, the biconditional is true. [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 16:52, 4 September 2025 (UTC)
::::: I disagree with GenAI. When I try &quot;{}↔{}&quot; in Gemini, it interprets it as &quot;The empty set exists↔The empty set exists&quot;. That makes no sense to me; in the language of first-order logic, &quot;{}&quot; is not a truth-value bearing expression, and it does not mean &quot;The empty set exists&quot;. This is not going to get us anywhere.
::::: In your latest page [[User:205.154.222.227/Theory_of_monism_panpsychism]], you start with &quot;assuming nothing&quot; (except for logic, but you do not explicitly say so there) and arrive at the conclusion &quot;Therefore energy is consciousness&quot; and &quot;Therefore consciousness is eternal, omnipresent, omnipotent, energy&quot;. That is another overt nonsense: you need some axioms of ''energy'' and ''consciousness''; you cannot derive anything about them purely from logic axioms.
::::: Above, I tried to supply additional dose of patience, especially for the benefit of the reader. I am inclined to no longer respond since this is not going anywhere and the nonsensical character of you page should be obvious to almost anyone who has the first idea of logic and philosophy. The people at https://atheistdiscussion.org/forums/showthread.php?tid=9242 agree with me; after having engaged with you a bit, they blocked you (and so did we at the English Wikiversity). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:08, 4 September 2025 (UTC)
::::: For the record, I have removed a fairly long post by the user, consisting mostly of favorable assessments of his material by various GenAI. Such material is already posted by him elsewhere (under a different user); no need to duplicate it here. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:48, 4 September 2025 (UTC)

For ease of reference, some locations where the debated/discussed material is available in some form (helpful search terms: &quot;nothing has the property of nothing&quot;, &quot;something has always existed everywhere&quot;, &quot;something is self-causal&quot;):
* [[User:MarsSterlingTurner/Consciousness]], 2025
* [[User:MarsSterlingTurner/Ontology]], 2025
* [[User:MarsSterlingTurner/What there is Consciousness]], 2025
* [[User:MarsSterlingTurner/What there is Consciousness/Monism-Panpsychism-Pantheism Theorem]], 2025
* [[User:MarsSterlingTurner/What there is Consciousness/Monism Panpsychism Pantheism Theorem/MarsSterlingTurner]], 2025
* [[User:205.154.222.227/Theory of monism_panpsychism]], 2025
* [[User:HumbleBeauty/Proof of monism]], deleted in 2020
* [[Wikibooks:User:HumbleBeauty/Proof of monism peer review]], 2020
* [[Wikibooks:User:HumbleBeauty/Worship technology]], 2020
* [[Wikibooks:User:HumbleBeauty/Proof of monism peer review2]], 2021
* [[Wikibooks:User:HumbleBeauty/Proof of monism peer review3]], 2021
* [[Wikibooks:User:MeekFavor/proofs]], 2023
* https://www.quora.com/Is-it-possible-to-reach-a-universal-truth
* https://www.quora.com/Have-all-scientific-attempts-to-date-failed-to-explain-why-there-is-something-rather-than-nothing-by-starting-with-the-assumed-existence-of-something
* [https://atheistdiscussion.org/forums/showthread.php?tid=9242 what pure logical tautologies can deduce], June 2025, atheistdiscussion.org
* https://ctmucommunity.org/wiki/User:HumbleBeauty/proofs
* https://ctmucommunity.org/wiki/User:HumbleBeauty/Theory_of_monism_panpsychism
* https://ctmucommunity.org/wiki/Proofs_of_God
* https://rationalwiki.org/wiki/Debate:Logical_proof_of_the_existence_of_God, 2014‎, user Selfreasoning4all
* https://rationalwiki.org/wiki/Debate:Metalogical_reasoning_for_the_Divine, 2014‎, user Selfreasoning4all
* https://onlinephilosophyclub.com/forums/viewtopic.php?t=3590, 2010
* https://forum.atheistrepublic.com/t/is-this-proof-of-universal-consciousness/6893/4
* https://www.ilovephilosophy.com/t/monopantheism/21176?page=2, 2008, user MySiddhi with identifying photo; people trying to explain why he was wrong back in 2008
* https://www.stormfront.org/forum/t1269204/, 2019, bumped in 2025 (warning: stormfront.org is some kind of white supremacist and anisemitic web)
* https://www.stormfront.org/forum/t1425039/, 25 June 2025, user SubtleBeauty; user Lord Jim provides some impressively patient and competent responses
My guess is that any additional page with similar content should be outright deleted rather than moved to user space; there are more than enough user space page where IP anons can edit and expand this kind of material. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:33, 8 September 2025 (UTC)
: Expanded. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:39, 9 September 2025 (UTC)
:: == Formal Proof of Consciousness; Introduction to Holomorphism Holographic Cybernetic Recursion ==
::This is a research project to investigate and public review the theory of proving universal consciousness or an eternal, omnipresent, all-powerful, self-causal thing; starting with assuming nothing while using set theory and scientific facts.
::(⊦∃{}') → ({}≡{}) → ({}={}) → ({}↔{}) → ({}→{}) → ({}:{}→{}) → ({}⊃{}) → ({}∻{})
::read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing; implies nothing equals nothing; implies nothing if and only if nothing; implies nothing implies nothing; implies nothing has the property of nothing; implies nothing is made of nothing; implies nothing is causal with nothing&quot;
::({}≡{}) → (∀x∃x) → (∃!x∀t∀s)
::read; &quot;nothing is nothing; implies for all things, they exist (nothing is nonexistence; nothing or nonexistence is not a thing; a thing is that which exists); implies there exists one thing for all time for all space (one thing is eternal and omnipresent and has the particular characteristics of existence)&quot;; note; without nothing or nonexistence existing, everything existing amounts to or generalizes to one thing!!!
::[(∃!x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)
::[(∃!x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s) → (∃x∻∃!x|∀t∀s)
::[(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)∧(∃x∻∃!x|∀t∀s)] → (∀W∈∃!x|∀t∀s)
::(∑E=Ek+Ep) → (∃!x(E)|∀t)
::read; &quot;the conservation of energy (energy cannot be created nor destroyed) implies one thing, energy, for all time (energy is eternal)&quot;
::(E=ħω/2) → (∃!x(E)|∀s)
::read; &quot;zero-point energy (there exists a particular amount of energy in every single point in space) implies one thing, energy, for all space (energy is omnipresent)&quot;
::[(∃!x(E)|∀t)∧(∃!x(E)|∀s)] → (∃!x(E)|∀t∀s)
::E = E
::∆E = ∆E
::∆E(∆s/∆t) = ∆E(∆s/∆t)
::∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)
::∆E = ∆E(∆s/∆t)(∆t/∆s)
::∆E/∆t = ∆E/∆s(∆s/∆t)
::∆W = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t
::∆W = ∇E∆v
::∫dW = ∫∇Edv
::W = ∫∇Edv
::[(∃!x(E)|∀t∀s)∧(W=∫∇Edv)] → (∀W∈E|∀t∀s)
::I have established logically that there is an eternal, omnipresent, all-powerful, and self-causal thing. I have shown as a scientific fact that energy has these qualities. According to the identity of indiscernibles, the said thing is energy!
::&quot;self-causal&quot; or (∃x∻∃!x|∀t∀s) may be consciousness of energy as every point in space has a holographic image or model of the universe; as &quot;every good regulator must contain or have access to a model of the system it regulates&quot;; https://en.wikipedia.org/wiki/Good_regulator_theorem (a theorem of cybernetics) the eternal, omnipresent, all-powerful energy, or the universe has a holographic model of the universe in every single point in space; as there is an omnidirectional converging and diverging flow of radiation into every single point in space; therefore energy is at least cybernetic according to the good regulator theorem.
::Language is isomorphic to reality; To describe reality requires language; Thinking/Describing transforms reality; Our consciousness is our reality; therefore consciousness is reality; therefore consciousness besides being self-causal or self-deterministic is also necessarily self-descriptive.
::Here is google AI formalized the proof of this fact;
::To formalize the proof that reality is self-descriptive and self-deterministic based on its isomorphism with language, we use a combination of '''Model Theory, Symbolic Logic,''' and '''Identity Logic.''' 
::1. '''The Isomorphism Premise''' 
::If reality (R) is isomorphic to language (L), there exists a structure-preserving, bijective mapping (f) between them. 
::Formalization: L ≅ R
::Meaning: Every state of affairs in reality has a unique, corresponding proposition in the language, and the logical structure of the language reflects the causal/ontological structure of the reality. 
::2. '''The Transformation Premise (Self-Reference)''' 
::The act of describing reality (D) is itself an event within reality (E ∊ R). If describing reality transforms it, then the language used to describe the system is a functional component of the system itself. 
::Formalization: D(L) → R' where R' is the transformed state of reality.
::Closing the Loop: Since L ≅ R, any operation on L is an operation on R. Thus, D(R) ∊ R. 
::3. '''Conclusion A: Reality is Self-Descriptive''' 
::If L is the only tool to describe R, and L is structurally identical to R, then R is its own meta-language. 
::Formal Logic: ∀x ∊ R, ∃φ ∊ L: f(φ) = x.
::Logical Identity: If L ≅ R, then the set of all descriptions of R (the &quot;theory&quot; of reality) is equivalent to the set of all states of R (the &quot;model&quot; of reality). This is the definition of a [[Self-Configuring Self-Processing Language]] (SCSPL). 
::4. '''Conclusion B: Reality is Self-Deterministic''' 
::Self-determinism follows because no external cause can exist outside of &quot;everything&quot; R to determine it. 
::Proof by Contradiction:
::Assume R is determined by an external cause C.
::By definition, R is the set of all that is real.
::If C determines R, C must be real.
::Therefore, C ∊ R. R is determined by a component within itself.
::Formalization: R = f(R). In a system where the &quot;rules&quot; (language) and the &quot;objects&quot; (reality) are isomorphic, the system's state at time T+1 is a function of its own internal linguistic/logical syntax at time T. 
::'''Summary of the Formal Proof Structure''' 
::Identity: L = R (as structures).
::Reflexivity: R contains its own description L ∊ R.
::Recursion: R processes itself through its own internal description (R → L → R').
::Closure: The system is &quot;syntactically closed,&quot; meaning it requires nothing outside itself to define or move its state, leading to Self-Determinism.
::Essentially, the holographic cybernetic model of energy is energy's theory of itself. Further evidence of panpsychism is found in the studies in parapsychological experiments; such as, Synthetic Telepathy; https://zerogeoengineering.com/wp-content/uploads/2025/04/From-Morse-Code-to-Mind-Melds-The-Rise-of-Synthetic-Telepathy-TFBT-April-2025.pdf , Telekinesis; https://web.archive.org/web/20200422213725/https://pearlab.icrl.org/pdfs/1997-correlations-random-binary-sequences-12-year-review.pdf https://icrl.org/articles-and-essays?from=pearlab , Telepathy; https://www.sheldrake.org/research/telepathy , Clairvoyance; https://web.archive.org/web/20220329004932/https://www.scientificexploration.org/docs/10/jse_10_1_utts.pdf , Precognition; https://www.youtube.com/watch?v=QSci5p5laSw , Biokinesis; https://tillerfoundation.org/media-resources/white-papers , Coherence; https://link.springer.com/article/10.1023/A:1006978911496 , and Correlations; https://noosphere.princeton.edu/ .
::Here is theoretical evidence of for example telepathy, clairvoyance, telekinesis, and biokinesis; IF quantum entanglement exists between separate brains it produces telepathy and IF quantum entanglement exists between brains and the outside environment then it produces clairvoyance and telekinesis and IF quantum entanglement exists between the brain and DNA and or RNA then it produces biokinesis; as spin can effect the thermodynamic properties of matter as proven through the spin isomers of hydrogen; https://en.wikipedia.org/wiki/Spin_isomers_of_hydrogen .
::Speculation hypothesis; the holographic model of the universe many be edited by the brain and produce parapsychological induced quantum entanglement between quantum recursion in the brain tubials via coherence brain wave radiation convergence and divergences in any point of space (theory/model/cybernetic universal hologram aspect) and therefore make the brain act like a transceiver of universal consciousness... and therefore edit reality. God's converging and diverging radiation in every single point in space is the holographic map. IT IS ALSO isomorphic to the territory. and is therefore self-descriptive. Quantum entanglement recursion would be the cybernetic regulator. Wherein holomorphism holographic cybernetic recursion is consciousness!
::This work was largely inspired by Wayne Dyer's Power of Intention; https://www.youtube.com/watch?v=Ct1dmO7y02E
::Fair well, and ps, to become innovative/add value to the universe/ via clairvoyance/ unity with the divine/ INTEND wellbeing for others. that is all you have to do!!! [[Special:Contributions/&amp;#126;2026-41028-15|&amp;#126;2026-41028-15]] ([[User talk:&amp;#126;2026-41028-15|talk]]) 22:07, 23 July 2026 (UTC)

== New Page Limit ==

I tried to respond on the talk page but my edit was blocked by the abuse filter (&quot;New User Exceeded New Page Limit&quot;). Could you please allow me to post my reply? [[User:Nazwa Shabrina|Nazwa Shabrina]] ([[User talk:Nazwa Shabrina|bicara]] • [[Special:Contributions/Nazwa Shabrina|kontrib]]) 12:20, 11 September 2025 (UTC)

:I want to fix the Sigma Spiral Constant page. [[User:Nazwa Shabrina|Nazwa Shabrina]] ([[User talk:Nazwa Shabrina|bicara]] • [[Special:Contributions/Nazwa Shabrina|kontrib]]) 12:22, 11 September 2025 (UTC)
:: For context, I tagged [[Sigma Spiral Constant]] for proposed deletion and placed a rationale/justification into the template.
:: You should have no difficulty editing the [[Sigma Spiral Constant]] page. As for editing [[Talk:Sigma Spiral Constant]], I would have thought you should be able to even as a new user. I just created [[Talk:Sigma Spiral Constant]] for the case that it would help you edit the talk  page. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:15, 12 September 2025 (UTC)
:::Now I can [[User:Nazwa Shabrina|Nazwa Shabrina]] ([[User talk:Nazwa Shabrina|bicara]] • [[Special:Contributions/Nazwa Shabrina|kontrib]]) 06:22, 12 September 2025 (UTC)

== Your proposed deletion of Basic Scratch Coding ==

Hello @[[User:Dan Polansky|Dan Polansky]],

I would like to inform you about your proposed deletion on the basic Scratch Coding page on Wikiversity. I am currently working on this project on Wikiversity and have created the project, and simply haven't found the time to contribute to the page, but I have managed to contribute to the page recently. I am currently improving the page, so I would like to remove the specific rationale.

Kind regards,

RailwayEnthusiast2025 —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 15:17, 15 September 2025 (UTC)

:P.S. : The Further Reading is on the 'Scratch' page, not on the 'basic Scratch Coding' page. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 15:18, 15 September 2025 (UTC)
:: A better venue for this topic is at [[Talk:Basic Scratch Coding]]. There I already provided some explanation. If that explanation is not enough or you have further questions, you can post there. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:20, 15 September 2025 (UTC)

== System- vs. User-Orientation ==

Why did you move the above from the main page to my user page. I am sick and tired.You may suppose it is my self-promotion. Then you are terribly.  It was my writing but ChatGPT's. No one could it write fairer than GPT.  Please geit it back. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 06:38, 25 September 2025 (UTC)
: 1) We are talking about [[User:KayYayPark/System- vs. User-Oriented Information Retrieval]]. There, we read &quot;Key contributions include: [...] K. Y. Park’s Direct Approach to Information Retrieval (1975), which emphasized user interaction with textual contexts (keywords and citations in context), representing an early user-oriented model.&quot; The page is not even marked as original research via a template and it does not state the author, thereby speaking in wiki voice. This is K.Y.Park (you) engaging in inappropriate promotion of a previously unpublished or poorly published thesis now available at [[A Direct Approach to Information Retrieval]].
: 2) As for &quot;No one could it write fairer than GPT&quot;, the idea that running something through ChatGPT (I am using Gemini) somehow consitutes a verification or validation (V&amp;V) is pretty absurd, except perhaps for a most superficial validation imaginable.
: As a result, I find my move fair and acceptable. My move is subject to review and reversal by full admins (or even semi-admins like myself?). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:44, 25 September 2025 (UTC)
: Even using Gemini does not validate your contribution, as per my new post here: [[User talk:KayYayPark/System- vs. User-Oriented Information Retrieval]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:47, 25 September 2025 (UTC)
: For the record: [[Wikiversity:Request_custodian_action#Block_of_User:KayYayPark]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:52, 25 September 2025 (UTC)

== Your opinion on Ninefold Resonance Theory ==

Dear Dan Polansky, I came across your interesting argument on [[Talk:May anyone call themselves a philosopher?]] and I was curious about your thoughts on my latest theory; the [[Ninefold Resonance Theory]]. If you would like to respond, please do so on [[Talk:Ninefold Resonance Theory|the discussion page]]! I look forward to hearing from you. Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 15:58, 28 September 2025 (UTC)
: I responded at [[Talk:Ninefold Resonance Theory]]. I fear you are going to be disappointed with my analysis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:16, 28 September 2025 (UTC)
::I didn't expect you to react with surprise and to see this as some kind of scientific breakthrough or anything, so I didn’t have high expectations! I really appreciate your feedback. As you can see, I have refined my theory a bit. I am curious to know if you think it's an improvement! [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 18:29, 28 September 2025 (UTC)

== Insertion of Photos ==

How can I insert private photos into the article? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 06:52, 29 September 2025 (UTC)
: You can upload photos (png?) on Commons (you have to be the copyright holder!). Home page: https://commons.wikimedia.org/wiki/Main_Page. Upload wizard: https://commons.wikimedia.org/wiki/Special:UploadWizard. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:53, 29 September 2025 (UTC)
::https://commons.wikimedia.org/wiki/File:B.C._Brookes_before_the_mailbox.jpg
::How can I  make the photo appear  from the above? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 07:37, 29 September 2025 (UTC)
::: Following the above link, I get &quot;No file by this name exists, but you can upload it.&quot; Your attempted upload must have failed? Other than that, the syntax &lt;nowiki&gt;[[File:B.C._Brookes_before_the_mailbox.jpg|thumb]]&lt;/nowiki&gt; would include the image on the page. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:39, 29 September 2025 (UTC)
::::Thank you so much. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 07:55, 29 September 2025 (UTC) 
::I've done it. Bother no more. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 07:54, 29 September 2025 (UTC)
::: You're welcome. I see, you uploaded this: [[:File:B.C. Brookes takes a motion to put a letter into the mailbox mouth.jpg]]. Although I think as a file, it is a bit too descriptive; it should ideally be shorter. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:00, 29 September 2025 (UTC)
::::Ya! I made it shorter. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 10:32, 29 September 2025 (UTC)
::::By the way, find my email adress from my &lt;nowiki&gt;[[User:KayYayPark]]&lt;/nowiki&gt; and email me anything so that I can know your email address. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 10:38, 29 September 2025 (UTC)
::::How stupid I am! My email address:  ishiakkum@google.com [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 12:12, 30 September 2025 (UTC)
::::: Your email address is a piece of sensitive, personally identifying information (it is one item that can, together with other items, help establish numerical identity of a human person). Revealing it on wiki is not necessarily what you want. Any bot can scan this page and then, someone can send you spam (even though you do have a spam filter, probably). You can ask your email address to be hidden (I think only a full admin can do it; I am a semi-admin/curator). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:16, 30 September 2025 (UTC)

== Peer review ==

I noticed with some alarm your combativeness with my deletion requets. On looking at your contributions, I see a substantial number of original research resources you have developed, few of which have been reviewed.

When are you going to seek peer review for them? How will you be arranging for such?

[[User:ජපස|ජපස]] ([[User talk:ජපස|discuss]] • [[Special:Contributions/ජපස|contribs]]) 16:11, 1 October 2025 (UTC)
: After I have seen your performance in another discussion in the English Wikiversity, I am inclined to disengage. I will respond to inquiries from Wikiversity custodians, for example. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:13, 1 October 2025 (UTC)
::Very well. I may start requesting many of your resources be deleted for lacking peer review. Apologies. [[User:ජපස|ජපස]] ([[User talk:ජපස|discuss]] • [[Special:Contributions/ජපස|contribs]]) 16:15, 1 October 2025 (UTC)
::: I am not aware of any policy-based requirement of a peer review for original research articles in the English Wikiversity. Nor has any custodian notified me of a requirement for peer review. Nor has, since the end of 2022, anyone notified me of such a requirement. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:17, 1 October 2025 (UTC)
::: Inspired by the above impulse, I asked myself where in the English Wikiversity one can find something like a peer review. I know of the following:
::: 1) [[Wikijournal]]s. They seem to take peer review quite seriously. That's something like a separate subproject of the English Wikiversity, so much so that there is a proposal to make it a completely standalone project.
::: 2) J.T.Neill is providing comments for his students in subpages of [[Motivation and emotion]]. However, it is not a ''peer review'', literaly speaking, since he is a teacher and not a peer/on the same level.
::: I cannot recall anything else. Perhaps there are more people like J.T.Neill; not that I remember right now.
::: I sometimes act as a partial reviewer by posting comments to pages, most recently here: [[Talk:Ninefold Resonance Theory]] (a page that is perhaps really for the user space rather than mainspace). But I assume no responsibility for comprehensiveness of these comments. (I have an extensive experience as a document reviewer from one of my previous jobs.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:52, 2 October 2025 (UTC)

== Clarification on authorship and licensing for Enhancing Web Browser Security through Cookie Encryption ==

Hello Dan,

I saw that you deleted and protected ''Enhancing Web Browser Security through Cookie Encryption'' citing possible copyvio from ResearchGate. I want to clarify that I am the original author of that ResearchGate publication. Since I hold the copyright, there is no violation. I am willing to release the text and figures under '''CC-BY-SA 4.0''', which is compatible with Wikiversity.

If needed, I can add an explicit license release statement on my userpage or the article’s talk page to confirm. The earlier move of the page was only due to a title mismatch (“Cookie Encryption”), not notability or copyvio.

Could you please advise how I should proceed to restore the page correctly with a clear license statement? I want to ensure the resource remains available under free licensing for Wikiversity readers.

Thank you for your time and guidance.

—Tom Joe James [[Special:Contributions/&amp;#126;2025-27242-55|&amp;#126;2025-27242-55]] ([[User talk:&amp;#126;2025-27242-55|talk]]) 15:56, 2 October 2025 (UTC)
: I checked https://www.researchgate.net/publication/391195563_Securing_and_Enhancing_Web_Browser_Security_through_Cookie_Encryption to see whether I can find any license statement. I find none. I cannot take a word from an anonymous IP account as reliable, in general; I have no means to verify the identity.
: But let's consider something else: what is the benefit from publishing the same text in the English Wikiversity? The text is already available as full text in researchgate.net, with references and properly formatted figures (which the page I deleted lacked). Surely the reader has all he needs in researchgate and suffers no loss from no article in the English Wikiversity? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:59, 2 October 2025 (UTC)
::The paper published in research gate is a pdf, which is not readable by LLMs or Web Crawlers properly.
::Publishing an Textual version of the paper ensures that the content is correctly indexed.
::And just like you, I've uploaded all the Wikiversity Rules to the GenAI and then asked for its opinions, it clearly stated that our paper is still eligible to be hosted in Wikiversity. (Source: ChatGPT &amp; Gemini after uploading Wikiversity several rule pages as Texts) [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 04:38, 3 October 2025 (UTC)
::: The link above is not a PDF (there is an additional button to get to a pdf). As for searchability by Google, I tried to search from  &quot;Securing browser cookies has been addressed by several&quot; (from the article) and it found researchgate and Wikisource. Meanwhile, the Wikisource page was deleted ([[S:Securing and Enhancing Web Browser Security through Cookie Encryption]]). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:44, 3 October 2025 (UTC)
::::Hello Dan, Kindly restore my previous comment that you have dismissed as &quot;GenAI&quot; Slop. So that others may follow our discussion. I have sent it so that you may understand my reasons. It was sent in a Good faith, not to spam or to vandalise.
::::The research gate link is is not pdf, but a document. Which cannot be viewed unless it is downloaded. There is a clear difference between a document and a web page. Even if the document gets indexed. It might not be read properly as it is structured differently. 
::::This is why a Wikiversity must be created for this document. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 04:54, 3 October 2025 (UTC)
::::: Anyone can see you post [[Special:Diff/2756895|here]]. I saw Wikipedia editors dismiss discussion input that is overt (and not signalled/indicated) GenAI slop. It seems a good idea to me to dismiss this kind of input. Sure enough, one can productively use GenAI to improve one's writing and discover items and ideas, but this overt voluminous slopping is a really bad idea, I think. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:57, 3 October 2025 (UTC)
::::::I used AI to expand my points to help you understand reasons behind my actions. Removing an entire reply from the main view instead of addressing it, is not a collaborative behaviour. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:16, 3 October 2025 (UTC)
::::: As for &quot;The research gate link is is not pdf, but a document. Which cannot be viewed unless it is downloaded.&quot;: That's not accurate: I viewed the document without downloading it (other than browser in a sense downloading any web page to serve it to the reader). The only problem with ResearchGate is that it seems to break down lines to achieve formatting so that Control + F on longer word sequences can fail. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:59, 3 October 2025 (UTC)
::::::Could you tell me how this paper is not eligible to be on Wikiversity?  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:18, 3 October 2025 (UTC)
::::::: Possible copyright violation, at a minimum. The April 2025 publishing venue (ResearchGate) has a page that does not indicate any license. Anyone responsible for the article would first have to contact ReserchGate and make them attach a license. Even then, we would need to clarify why ''double publication'' is an acceptable thing in this case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:24, 3 October 2025 (UTC)
::::::::I will add the license as soon as possible in the researchgate page.
::::::::And this is not a double publication, Research Gate is a self archive. And publishing in Wikiversity ensures accessibility. Improving visibility and indexing. We do not want our researchgate profile to be indexed. We want the content inside the paper to be indexed so that in the future LLMs can index it and train on the data. Research Gate acts as a formal preprint.
::::::::I hope everything is clarified.[[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:01, 3 October 2025 (UTC)
::::::::: Disclaimer: I am inexperienced with this matter. I make no promises on what will be allowed in Wikiversity. There are other admins (I am a semi-admin) around, and they can help resolve the relevant questions, I think. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:40, 3 October 2025 (UTC)
: Moreover, I asked Google Gemini: &quot;In Academic publishing, is it okay to publish the same article in multiple venues?&quot; It said: &quot;It is generally not okay to publish the same exact article in multiple academic venues. This practice is considered unethical and is known as duplicate publication or redundant publication.  [...]&quot;, with two links to sources. Sure enough, I went the cheap way and GenAI cannot be considered reliable, and would need to go to the sources and investigate the issue. But as a first quick impression, it reinforces my decision not to support recreation of the article in the English Wikiversity until someone can convincingly argue that GenAI is wrong. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:12, 2 October 2025 (UTC)
: For the record, I removed from this place a post that is a GenAI slop (see [[Special:Diff/2756895|this edit]] from [[User:Tomlovesfar]]). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:39, 3 October 2025 (UTC)
: (I am now responding here at the bottom, in part to a subthread above, to make it easier.)
: The principle implied in the proposed action is this:
:: &quot;The English Wikiversity should allow itself to be used as a double/duplicate repository for appropriately licensed content from ResearchGate, in part to drive LLMs.&quot;
: I am not sure/clear that this is a good principle, for the English Wikiversity. (Greetings to Kant, so hated by Rand.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 3 October 2025 (UTC)
::I’d like to clarify that this is '''not''' a case of double or duplicate publication in the academic sense.
::'''ResearchGate''' is a '''self-archiving platform''', similar to a preprint repository. It is not a peer-reviewed journal, and its content is not indexed or presented in a web-friendly, accessible format.
::'''Wikiversity''', on the other hand, is an open educational platform, with the goal of making knowledge freely reusable, remixable, and indexable (especially in plain-text formats). Publishing here is not a &quot;duplicate&quot;, but a '''different mode of dissemination''' serving different audiences.
::While driving LLM accessibility is a modern consideration, the '''primary goal is broader accessibility''' for human learners, researchers, and educators who may not access or index ResearchGate content easily.
::The version proposed for Wikiversity is '''freely licensed, educational, and structured for reuse'''
::As for Kant and Rand, I’ll just say I’m aiming for pragmatic clarity, not dogmatic idealism [[Special:Contributions/&amp;#126;2025-27507-81|&amp;#126;2025-27507-81]] ([[User talk:&amp;#126;2025-27507-81|talk]]) 07:09, 3 October 2025 (UTC)
::: That's it. I am not engaging with GenAI slop. I am keeping the above slop for the reader; I think I will remove any other slop. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:20, 3 October 2025 (UTC)
: As an item of public deliberation: we do not know whether the above interaction is with someone named &quot;Tom Joe James&quot;; neither self-disclosure by an anonymous IP editor nor relevantly named user account firmly establish the matter/the identity. It could well be just a disruption from an entity that for some reason wants to disrupt the English Wikiversity (why would they target such backwaters is not clear, but they could). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:02, 3 October 2025 (UTC)
::The IP addresses belongs to Tom Joe James. 
::I have been editing from my mobile phone, hence the guest edits.
::All of the above comments and replies were from me. You can check out my User:tomlovesfar and I have been contributing to the English Wikipedia under the same username tomlovesfar. 
::However, after updating my license on the research gate page, I would proceed to recreate the page under the same title but from my main account. That will be all from my side, Kindly do not delete any of the pages without proper discussion or warnings. 
::Best Regards, 
::Tom Joe James [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 08:15, 3 October 2025 (UTC)
::: Unless you send me a scan of your ID document, I have no way of reliably knowing who you are (but even ID documents can be fake). And as a curator/semi-admin, I guess I am not trusted to deal with this kind of sensitive information.
::: I guess you better chance is with other curators and custodians than with me. You can ask someone to review my approach. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:18, 3 October 2025 (UTC)
::::That isn't necessary, you can head over to my LinkedIn and under volunteering section I have mentioned my wikipedia contributions along with my username. 
::::or checkout my orcid id - 0009-0000-4896-9046
::::&lt;nowiki&gt;https://orcid.org/0009-0000-4896-9046&lt;/nowiki&gt;
::::That is more enough to prove it. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 08:25, 3 October 2025 (UTC)
::::: So I checked https://www.linkedin.com/in/tomjoejames-/; I do not see Wikipedia user account stated (&quot;Tomlovesfar&quot;)? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:30, 3 October 2025 (UTC)
::::::my apologies, I forgot that I had removed the screenshots. 
::::::I have updated it once again. Kindly check it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 08:35, 3 October 2025 (UTC)
::::::: There are screenshots, but a search for &quot;Tomlovesfar&quot; does not find anything ''in the text''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:36, 3 October 2025 (UTC)
::::::::Yes, I have updated it in the description.
::::::::&quot;User:tomlovesfar&quot; [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 08:39, 3 October 2025 (UTC)
::::::::: Okay, now we know that some entity has access both to the above IP account and to the LinkedIn account (really? am I jumping to conclusions?).
::::::::: I propose the following: 1) you make sure ResearchGate text has proper license there; 2) after that is completed, you contact some custodians (I am a curator) to review my approach. Before 1) is completed, any further communication is beside the point. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 3 October 2025 (UTC)
::::::::::Hello Dan,
::::::::::As requested, I have updated my ResearchGate publication with an explicit '''CC BY-SA 4.0 license statement'''. I’ve also mirrored this on my Wikiversity userpage for verification.
::::::::::Could you please reconsider restoring the article, or alternatively allow a custodian to review?
::::::::::Thank you,
::::::::::—Tom Joe James [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:14, 3 October 2025 (UTC)
: (Outdent) I skimmed the article at https://www.researchgate.net/publication/391195563_Securing_and_Enhancing_Web_Browser_Security_through_Cookie_Encryption. I also searched (Control+F) for &quot;Licen&quot;. I did not find anything. Where in the page should I look? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:16, 3 October 2025 (UTC)
::Securing and Enhancing Web Browser Security through Cookie Encryption
::* April 2025
::* DOI: 10.5281/zenodo.15287972License
::* CC BY-SA 4.0 you should find it under the Title
::[[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:18, 3 October 2025 (UTC)
::: I don't see the license item. I refreshed the browser as well, also hard refresh using Control+F5. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:20, 3 October 2025 (UTC)
::::[[File:Snapshot of a research gate article preprint.png|thumb]]
::::I have opened the link using incognito, It is still visible to me. 
::::Kindly Check this screenshot as well as this link. if possible send me your screenshot
::::https://www.researchgate.net/publication/391195563_Securing_and_Enhancing_Web_Browser_Security_through_Cookie_Encryption
::::You can find it under the date &amp; DOI number [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:25, 3 October 2025 (UTC)
::::: Still not seeing it; I tried a browser I hardly ever use, but I do not see it there either. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:27, 3 October 2025 (UTC)
::::: I also do not see the yellowish/orangish rectangle about ''preprints''. Perhaps you page is in ''preprint mode'' so I cannot see the change? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:39, 3 October 2025 (UTC)
::::::I think that you being served an old cache version. Maybe logging in would serve you the latest version. Or try clearing cache   [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:53, 3 October 2025 (UTC)
::::::https://zenodo.org/records/15287972
::::::Here is the Zenodo Link, Where our DOI is registered. The License is clearly mentioned under the tab &quot;Rights&quot;
::::::Hope this helps, Let me know how we can further proceed. [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 17:55, 3 October 2025 (UTC)
: (Outdent) I am still not seeing the license, and as I said, I used a different browser so the problem should not be in the browser cache.
: Moreover, I started [[Wikiversity:Colloquium#Publishing text from ResearchGate in Wikiversity as a copy]] to collect input from others. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:09, 3 October 2025 (UTC)
::I do not understand how this is possible. I have updated it in Zenodo, It should be visible to you.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 02:44, 4 October 2025 (UTC)

== Discussion regarding your behaviour as a curator ==

Hello, Just letting you know that I’ve raised a concern about our recent interaction at the [[Wikiversity:Colloquium#Concern regarding curator conduct User:Dan Polansky|Colloquium]] for review by other custodians. 

This is not meant to escalate conflict but to ensure transparency and get a fair, neutral perspective on how the discussion unfolded. [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 18:57, 6 October 2025 (UTC)

:I second that [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:03, 15 October 2025 (UTC)

== Preview ==

Do you use preview? It would be easier to look to the history, then if you use it a do not save with multiple small edits in a short period of time. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:11, 6 October 2025 (UTC)
: I follow an editing process that I find most productive for idea capture, which involves many rather small edits to my sandbox. I am investigating alternative options, but this is extremely productive. From my experience, editing in a long series of small edits is fairly common in the English Wikiversity, so I do not expect objections from native English speakers. As for the ease of reviewing the editing history, there is no need for anyone to inspect the editing history of my sandbox. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 20:15, 6 October 2025 (UTC)
::Your talk page is your sandobox? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:23, 6 October 2025 (UTC)
::: I thought you were referring to my editing in small edits of [[User:Dan Polansky/sandbox]], which is very extensive to the point that I am exploring alternatives (but it is very productive!) As far as I recall, my editing of my talk page does not suffer from this that much. It rather often happens that I save a response and only then realize I would like to amend it. So I do amend it. In the end, my posts are sometimes made in 2-3 edits. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 20:28, 6 October 2025 (UTC)
::::I see. Well, but that may lead to edit conflicts. I usually tend not to fix grammar erors send to the server not to cause such problems. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:36, 6 October 2025 (UTC)
::: As I was thinking about it, the above &quot;Your talk page is your sandobox?&quot; shows overt carelessness: 1) &quot;sandobox&quot; should not have &quot;o&quot;; 2) it should have a proper English question sentence form, &quot;Is your talk page your sandbox?&quot; But even that sounds sort of irritating to me; something is off. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:04, 8 October 2025 (UTC)
: (Outdent) Some questions to be clarified:
: 1) When editing one's talk page or another page where one participates on a discussion, should one carefully review the text before posting it, to avoid edit conflicts?
: 2) When (in relation to the above), even with best care, one notices a need to edit one's post to a discussion, and when no one has responded yet, should one avoid further editing one's post, in view that this may lead to an edit conflict, causing a complication for the responding person?
: 3) If the answer to item 2) is no, one should take risk of edit conflict: should the person suspecting and fearing an edit conflict edit their response e.g. in Notepad and avoid posting a response until some reasonable time elapses, e.g. 5 minutes?
: 4) When one is not a native speaker of the language of communication (here English), should one remind oneself that one is composing in a foreign language, which will naturally lead to more mistakes that need correcting? Should one therefore be tolerant/kind toward oneself, and accept the repeated edits as a cost of trying to deliver value in a foreign language?
: --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:49, 8 October 2025 (UTC)

== Seperate See Also Page  ==

Why can the seperate '''[[See Also]]''' page not be transplanted into another page with &quot;{{See_Also}}&quot;? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 08:05, 10 October 2025 (UTC)
: Instead of responding to the above, let me point out I recently opened a request: [[Wikiversity:Request custodian action#Block of User:KayYayPark or prohibition of new pages]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:13, 10 October 2025 (UTC)
::Why do you like me to be blocked? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 15:01, 10 October 2025 (UTC)
: Let me note that it is quite remarkable that you apparently forgot the skills that you had under [[User:KYPark]] account. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:52, 10 October 2025 (UTC)
::Yes, I am supposed to have lost a lot. What a pity! It seems so because of my old age. The above question is quite embarassing, because I confirmed it from my old page. There is nothing wrong with my present practice. Anyway you must know what's wrong with &quot;See Also&quot; of [[Memex]]. Please tell me. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 15:08, 10 October 2025 (UTC)
::There, &lt;nowiki&gt;{{See_Also}}&lt;/nowiki&gt; returns Template:See Also in red color. Why not returning properly? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 15:16, 10 October 2025 (UTC)

== Regarding Wikiversity:Colloquium ==

Hello Dan,
After looking at https://en.wikiversity.org/wiki/Wikiversity:Colloquium#Concern_regarding_curator_conduct_User:Dan_Polansky and my own personal experiences, I would like to politely ask you to instead of asking for deletion of my learning resources, try and help me improve and give advice. After all, I am fairly inexperienced at Wikiversity. I understand that you might be helping at Wikiversity, but for similar situations with this deletion matter, please hand it over to other curators or custodians such as @[[User:Atcovi|Atcovi]] or @[[User:Koavf|Koavf]]. In the meantime, please can you give me advice on how to improve, as I have seen you are fairly experienced here.

Thanks, 

RE —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 20:16, 13 October 2025 (UTC)

:As an immediate concern, if there is some learning resource that in principle belongs here but in practice is insufficient or inappropriate, userfying it is a good short-term solution. Just move a page from ''foo'' to ''User:RailwayEnthusiast2025/foo'' and build it up until it's acceptable and move it into the main namespace. I'm not knowledgeable enough about any particular resource or making any implicit claim about the quality of your work, just saying that this is a perfectly amenable solution. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:26, 13 October 2025 (UTC)
:: @RailwayEnthusiast2025: In [[Special:Diff/2745947|this edit]] from 18 September 2025, you removed request for deletion from [[Basic Scratch Coding]], which was really a request for ''moving to user space''. Rather than ''delete'' pages, I usually move them to user space, based on Wikiversity tradition; an exception is when there is a suspicion of copyright violation, but even in such cases, I plan to use [[Template:Copyrighted]] much more, in case of doubt, instead of immediate deletion.
:: As it is, [[Basic Scratch Coding]] is not ready for mainspace. Ideally, it should be moved to user space, after which you will have ''indefinite time'' to work on it.
:: A relevant discussion already took place at [[Talk:Basic Scratch Coding]], where I said: &quot;I now added a proposed deletion since in this state, the course is of little use. There is a protective period of 3 months before deletion, expiring on December 14, 2025. Even then, the page would typically be not outright deleted but rather moved to user space. I feel this is not too harsh; by Dec 14 2025, over a year will have elapsed from this page having been started&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 21:02, 13 October 2025 (UTC)
:: A note, perhaps an aside: I got involved with pages by Harold Foppele after he pinged me at [[Talk:Quantum A Matter Of Size]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:27, 13 October 2025 (UTC)
:::What have i to do with all this? DFX [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:10, 15 October 2025 (UTC)
:::Actually, you are correct. If a project is in mainspace, it should help participants learn. Draft namespace and userspace is a place in where you can develop your pages. Thanks for helping me understand. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:50, 6 November 2025 (UTC)
:::: Thank you. Let me add that I am not the boss here: if I assess a page as not fit for the mainspace, you or anyone else can challenge my decision at [[WV:RFD]] as a request for undeletion (moving out of mainspace is quasi-deletion) and then we call collect input from others on your artifact (page or a set of pages). I am fallible and make mistakes, more than I care to admit, but I am trying to learn from them, again and again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 19:11, 6 November 2025 (UTC)
::So creation of a learning materials is no longer community driven? Why they cannot be developed in the main name space? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:35, 16 October 2025 (UTC)
:::I don't know. I only really moved these pages to my user space due to @[[User:Dan Polansky|Dan Polansky]] giving the advice that my projects weren't good enough for &lt;s&gt;my namespace&lt;/s&gt; mainspace. &lt;s&gt;It &lt;/s&gt; His language seemed as if he were to drive me off Wikiversity and as if my contributions weren't valued. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 16:33, 16 October 2025 (UTC)
::::*mainspace —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 16:34, 16 October 2025 (UTC)
::::My question was for Justin, as you can see I am idnenting it below his post. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:02, 16 October 2025 (UTC)

== Is it you? ==

Is it you who has deleted [[Korean/문맥색인요약]], and some other Korean articles under [[Korean]]? Then please explain why these are unfair to be there under [[Korean]]? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 13:29, 16 October 2025 (UTC)

:You can [https://en.wikiversity.org/wiki/Korean/%EB%AC%B8%EB%A7%A5%EC%83%89%EC%9D%B8%EC%9A%94%EC%95%BD?action=edit&amp;veswitched=1 see here] @[[User:KayYayPark|KayYayPark]], who have deleted that page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:49, 16 October 2025 (UTC)

:: Thank you but I am sorry the above link links to the deleted page. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 08:26, 17 October 2025 (UTC)

: The page was deleted by [[User:MathXplore|MathXplore]] with the summary &quot;Not English. See https://www.wikiversity.org to find where contributions may be appreciated&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:11, 3 November 2025 (UTC)

== Questions about qualifications and questioning expertise ==

Do you think it is appropriate to [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&amp;oldid=2760143#Your_qualification ask users what their qualifications are] and [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&amp;oldid=2760190#Contest_removal_of_article question their qualifications]? Not me. Wikiversity is not some kind of professional workplace where we would copy the structures of a real university and decide who has what degrees or expertise. Wikiversity is an open educational space that allows even those who are not qualified to teach and learn. The question is also how you can recognize the qualification in the field of physics when you yourself write that [[User:Dan Polansky|you are a programmer]]. But even if you could, there are certainly more appropriate ways dispute the quality of the resource, as the discusion on the resource talk page or [[:Template:Citation needed]]. If you are unhappy with these traditional or available ways you may propose new policy on such a content. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 16 October 2025 (UTC)
: The policy proposal [[Wikiversity:Artificial intelligence]] states:
:: &quot;Generative artificial intelligence large language models may be able to generate text that may be suitable for inclusion on Wikiversity. However, to be acceptable:
::: &quot;The contributor should be an expert on the topic
::: &quot;[...]&quot;
: I suspected that the user used GenAI. I therefore found it appropriate to investigate whether he was an expert on the topic of quantum mechanics, or at least had a half-decent basic knowledge. Out of courtesy, I expressly indicated that a response to my inquiry was not required by a policy (since this is a policy ''proposal'', not a policy): &quot;Since you are entering articles on advanced physics, it would be worthwhile to know your qualification, e.g. university-level degree. Disclaimer: as far as I know, Wikiversity has not introduced any codified requirement of your disclosing such qualification.&quot;
: If the English Wikiversity determines that such an inquiry is undesirable/inappropriate, I can accept that, even though somewhat begrudgingly, since I think a ''voluntary'' disclosure of one's qualification is a very good thing in the context of the English Wikiversity (and should even be ''mandatory'' for those who insert GenAI output?).
: On a relational note, I do not think you are in the position to make edicts telling me what I may and may not do; you can make proposals at best. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:00, 3 November 2025 (UTC)
::It has been [[Wikiversity:Etiquette|determined here]] since 2006. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 17 November 2025 (UTC)
::: The page [[Wikiversity:Etiquette]] has neither policy nor guideline status. The consensual support for that page is unclear. Whether anyone carefully reviewed the page with the intent of finding errors and possible harm resulting from the page is unclear. For instance, the page says &quot;Use edit summaries&quot;, but that is very often ignored in the English Wikiversity; this suggests that the page as a whole does not have consensus. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:40, 18 November 2025 (UTC)

== Testing newbies is not civil ==

I am sorry Dan. You are not here in the position of an authority [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&amp;oldid=2759624 to test newbies]. You can dispute content - if you see copyright violation you can ask for the prove, that the author uses its text or delete it. But why you investigate the users this way? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:01, 16 October 2025 (UTC)
: My response above at [[#Questions about qualifications and questioning expertise]] is applicable here. By asking &quot;Can you, in your own words, quickly describe the double slit experiment here?&quot;, I tried to determine whether the user had at least a very basic idea of quantum mechanics, in part since that was required by the policy proposal, and in part since it has bearing on ''authorship'' of content being entered into Wikiversity.
: The link you provided, https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&amp;oldid=2759624, shows the editor to be very problematic, yet you take his side and take no reservation to his conduct. (Part of the response self-censored as perhaps better avoided, after all.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:45, 3 November 2025 (UTC)

== [[Korean/게르만어와한국어]] ==

Is it you who deleted the above page? --[[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 11:20, 22 October 2025 (UTC)

This was edited directly so that I have no copy. I was going to upload it to x.com. Please do me a favor to return that deleted copy back to me either way, say, via email ishiakkum@gmail.com.  [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 11:46, 22 October 2025 (UTC)
: Email sent. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:38, 22 October 2025 (UTC)
: The page was deleted by [[User:MathXplore|MathXplore]] with the deletion summary &quot;Not English. See https://www.wikiversity.org to find where contributions may be appreciated&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:05, 3 November 2025 (UTC)

== MediaWiki questions and answers ==

Moving the issue to the [[Wikiversity:Colloquium#MediaWiki questions and answers|Coloquium]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:17, 26 October 2025 (UTC)
: The above is in reference to [[User talk:Juandev#Page move]], September 2025, where I spent quite a bit of effort to explain why keeping [[User:Juandev/MediaWiki questions and answers]] in mainspace is a bad idea.
: You can request undeletion of pages (move to user space is quasi-deletion) in [[WV:Requests for deletion]]; that is the standard process. On the other hand, using Colloquium is perhaps okay, especially if you want to clarify ''general principles''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:08, 3 November 2025 (UTC)

== Community Review ==

Please see [[Wikiversity:Community Review/Dan Polansky]]. I have temporarily removed Curator rights pending the outcome of the discussion. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 06:22, 16 November 2025 (UTC)
: @[[User:Mu301]]: I responded there. Can you clarify why you removed the rights before the discussion rather than after the discussion? Did I misuse the rights in any way, or was there a risk of my misuse of the rights? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:55, 16 November 2025 (UTC)

== Night mode unaware.. ==

Hitting a limit as to what I can usefully patch at my access level. 

I've sandboxed a new version of Collapse top. ([[Template:Collapse top/sandbox]] Please review. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:22, 17 November 2025 (UTC)

: The remaining non content or userspace instances of the Lint concerned are mostly template transclusions, which need the protected templates to be patched :(. I can't do this.  so would appreciate someone with the relevant access doing so, based on the approach I have been taking. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:22, 17 November 2025 (UTC)
:: I am not clear what I can do for you; I was a curator, and currently, I do not have the curator rights (see also [[Wikiversity:Community Review/Dan Polansky]]). So either I can do that for you if and after I get the curator flag back, or perhaps you are better off going to Colloquium. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:25, 17 November 2025 (UTC)

== Trolls ==

Do you know that it's not appropriate to publicly call someone a troll? It only leads to an escalation of the conflict. If someone bothers you, it doesn't mean that they are a troll. It's just a personal incompatibility. In order for someone to be called a troll, there must be a consensus of several users, and the discussion about it often takes place offline to avoid developing a conflict. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:50, 17 November 2025 (UTC)
: I am not clear that engaging the above multi-wiki convicted troll (intentional disruptor causing significant harm/economic loss) Juandev is a good idea (more at [[Wikiversity:Community Review/Dan Polansky]]). For anyone reading this and wanting to know about the topic of troll-labeling, I will write something.
: My understanding is quite different. The above makes it seem as if there is some universally accepted and universally valid principle unconditionally prohibiting troll-labeling. But I have seen a different practice. I have myself said of user AP295 that I wondered whether his behavior was a kind of trolling and no one complained ([[User talk:AP295]], &quot;I wonder whether what I am seeing is a certain kind of trolling.&quot;). AP295 was indeffed in the English Wiktionary with reference to trolling, and the block of AP295 at Meta also referenced trolling, per https://meta.wikimedia.org/w/index.php?title=Special:CentralAuth&amp;target=AP295. Disregarding now the observable actual practice, early detection and neutralization of intentionally disruptive editors is vital for the health of the project, as the English Wikiversity should know by now only too well, with memory of major disruptors Abd and Marshallsumter. One may perhaps wonder whether the label &quot;disruptor&quot; is better than &quot;troll&quot; since it does not impute intent, but in the case of Harold Foppele, the disruption being intentional is manifest. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:34, 18 November 2025 (UTC)

== 1-week block ==

See [https://en.wikiversity.org/w/index.php?title=Wikiversity:Community_Review/Dan_Polansky&amp;diff=prev&amp;oldid=2775054 here] for the reasons. Inappropriate usage of your talk page will also result in you losing access to it. 

Also, your [[User:Dan Polansky/User Juandev|page on Juandev]] is not welcomed here, and per your comment: &quot;If some custodian considers this manner of collection inappropriate, I can accept that (even if begrudgingly) and I request that the content of the page is sent to my email if the page is deleted&quot;, I will send the content of this page to your email so you can have it for your ''own'' records and have it be off-wiki. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:09, 21 November 2025 (UTC)
: I will copy your reason here for ease of reference, from [[Special:Diff/2775054]]: &quot;I have blocked Dan for 1 week due to the obvious disruptive conduct taking place in this community review. Dan is using this community review as a way to launch unsupported attacks against other users, including [[User_talk:Atcovi#Child_psychology/Summary_of_child_psychology_(cheat-sheet)|allegations of me using &quot;GenAI&quot; to create my pages]] and an [https://en.wikiversity.org/w/index.php?title=Wikiversity:Community_Review/Dan_Polansky&amp;diff=prev&amp;oldid=2775006 unnecessary comment on another user's contributions to our project]. Also, the [[WV:Canvassing|canvassing]] taking place here is also indicative of Dan's lack of remorse for his actions. Lastly, the LTA, who is also asserting Dan's racist comments (which Dan has not addressed nor expressed remorse for), leaves a bad impression on our community. I want to remind everyone that [https://meta.wikimedia.org/wiki/Requests_for_comment/Shut_down_Wikiversity proposals for this project's removal have took place in the past], and we surely don't want another discussion to be brought up for WV's removal. I'm hoping that the block imposed here will give some time for the community to productively discuss this community review, and for Mike to make his comments or give his final verdict. If Dan returns after the block and continues to pursue his disruptive, slanderous campaign, a harsher block will be imposed. Thanks.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:25, 21 November 2025 (UTC)
: Can you clarify which canvassing are you talking about? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:25, 21 November 2025 (UTC)
: Can you explain why you deleted [[User:Dan Polansky/User Juandev]]? The page contained detailed evidence supporting a measure agains Juandev. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:26, 21 November 2025 (UTC)
: As for your statements that I am making allegations, I have ''asked a question'', not alleged: &quot;Since the pages have at least a modicum of GenAI wibe, I would like to ask you whether you used GenAI to compose these pages, whether in whole or in part. The policy draft/policy proposal concerning GenAI is WV:Artificial intelligence.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:33, 21 November 2025 (UTC)
: As for &quot;Dan's racist comments&quot;: my task as the wikidetate/pro-con-analysis creator was to provide arguments for both sides rather than only arguments I agree with. I therefore provided the most salients arguments I could figure out that were in the air, as it were. These were not ''my'' comments; these were comments of an abstract potential Wikidebate participant, something like a literary figure. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:39, 21 November 2025 (UTC)
:: As per [[Wikidebate]], &quot;Wikidebate is a collaborative learning project to gather and organize ''all'' arguments on controversial issues&quot;, italics mine. Granted, the word &quot;all&quot; needs to be interpreted somewhat non-literarily; the arguments need to have a modicum of relevance and strength. But it is not suggestive of censorship, not even self-censorship. You did not yet take the opportunity to open a Colloquium discussion about [[Wikidebate]] and the debate format; why not? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:10, 21 November 2025 (UTC)
: Okay, I noticed [[Special:Diff/2775076|your another edit]] where you seem to indicate that the following is a &quot;canvassing&quot;. Is that right, is that canvassing? &quot;As an aside, can you help support my proposal at [[Wikiversity:Colloquium#General ban on direct use of GenAI output with exceptions]] so that we can curb/limit a developing harm of the English Wikiversity, the mounting volume of mainspace pages with predominantly GenAI-generated output?&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:47, 21 November 2025 (UTC)
: I would like to remind you that the blocking tools include the option to block specifically [[Wikiversity:Community Review/Dan Polansky]], if that is the main part of rationale. If you change the block to that page specifically, or also to mainspace, I will be able to continue post my analyses to my user space, for the benefit of myself and perhaps others. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:32, 21 November 2025 (UTC)

== Three-month block ==
I was blocked by user Atcovi for the following: &quot;continued disruptive editing, unwillingness to cooperate, cross-wiki harassment, and further time-wasting&quot;.

I do not know what that refers to. No diffs or links were provided. I consider the unsubstatiated statements and the block itself highly inappropriate. I find it libelous. It is not acceptable to talk in generalities like that without a shred of substantiation, to my mind.

The overt disruptor Harold Foppele continues unabated, with no block against him at all. This is all just absurd.

Today, I opened the following discussion, in which Atcovi should provide answers:
* [[Wikiversity:Colloquium#PSYC322 - Adolescent Psychology and potential plagiarism]]
The discussion is about pages by Atcovi and his ''potential'' plagiarism. His use of block tool against me after this is, to my mind, highly inappropriate.
What has become of the English Wikiversity? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:57, 17 December 2025 (UTC)

== Request to Jtneill ==
@[[User:Jtneill]]: you were my mentor when I was a curator. Let me ask: is the indefinite block of me appropriate? Why is not a one-year-block more than enough? Did I really ''disrupt'' Wikiversity or did I rather do a lot to help it (keyword: Marshallsumter)? Are there any uses of my curator tools that were determined to be inappropriate? Thank you and sorry to bother you. I am quite distressed.

(I also requested this back in November 2025: [[Wikiversity:Request custodian action/Archive/29#Indefinite block for Harold Foppele]]. Maybe I was too clumsy or inexpert in that request; maybe the substantiation was not good enough. The user account is now globally locked. The cleanup after this disruptive account in the English Wikiversity is still pending, I think; by my assessment, all pages he created should be moved out of mainspace to his user space.

It was also in November when I requested this: [[Wikiversity:Colloquium/archives/November 2025#General ban on direct use of GenAI output with exceptions]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:30, 10 March 2026 (UTC)

@[[User:Prototyperspective]]: You just pinged me, but I am indefinitely blocked since today. I am not going to respond on substance since that could be seen as a block evasion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:37, 10 March 2026 (UTC)

On the GenAI note, let me note that adopting [[Wikiversity:Artificial intelligence]] as is is ''super unwise''; it is a de facto go head to placing GenAI output in mainspace in volume. While there is &quot;The text should ideally be revised and rewritten by the contributor&quot;, the word &quot;ideally&quot; makes it optional. There is now a risk that this bad proposal is going to be adopted, as per [[Wikiversity:Colloquium#Wikiversity:Artificial intelligence to become an official policy]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:53, 10 March 2026 (UTC)

:Consensus has been reached to block you indefinitely per the Community Review. Your continued lack of remorse and abuse of your talk page privileges has led you to this point. I earnestly suggest you take this time to reflect on your actions and way of handling high-stake situations, and I hope you return to our project in the near future with these issues recitified after a successful appeal. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 23:10, 10 March 2026 (UTC)

:{{ping|Dan Polansky}} Sorry, I haven't been around much over last few months. It seems that your strong views and associated actions, whilst well intended, caused too much friction and led to consensus to block. Perhaps there are some lessons here that, if learnt, could lead to undoing the indefinite block. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:03, 24 March 2026 (UTC)</text>
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      <comment>Reverted edit by [[Special:Contributions/~2026-41028-15|~2026-41028-15]] ([[User_talk:~2026-41028-15|talk]]) to last version by [[User:Jtneill|Jtneill]] using [[Wikiversity:Rollback|rollback]]</comment>
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== Could you do another manual revert on [[Web design]] now that it’s semi-protected? ==

See https://en.wikiversity.org/w/index.php?title=Web_design&amp;diff=next&amp;oldid=1936739 (the [[Special:Contributions/2402:8100:3009:2DEE:1:2:F528:AAC9|only contribution from that IP]]). They replaced the CSS Zen Garden screenshot – which is admittedly somewhat outdated, but still more relevant to web design than a 19th century painting.
--[[Special:Contributions/78.23.192.69|78.23.192.69]] ([[User talk:78.23.192.69|discuss]]) 21:16, 29 March 2025 (UTC)
: Reverted as you proposed ([[Special:Diff/2710640|diff]]}}. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:48, 3 April 2025 (UTC)

== What prevents Monism Panpsychism Pantheism Theorem from being posted on wikiphilophers? ==

Hello, thank you for all your replies so far.

But I have responded to everything you made challenge of. I removed the empty set, I removed nothing implies nothing... and I rebutted your claim that nothing could logically exist.

Do you have any more challenges to my proof?

If not, could you add my proof to wikiphilosophers?

https://en.wikiversity.org/wiki/User:MarsSterlingTurner/What_there_is_Consciousness

I know you're an atheist. But that should not stop me from adding a proof of this quality from wikiphilosophers.... should it?

Only Christopher Langan's CTMU and Baroch Spinoza's Ethics come close to the quality of the proof that I provided... and both are considered academic material. [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 00:24, 9 April 2025 (UTC)
: [[User:MarsSterlingTurner/What there is Consciousness]] is rank nonsense.
: The question is at what length one should spend time here and effort to articulate point-by-point that your nonsense is nonsense; I don't know.
: Your user page has some red flags as well, e.g. &quot;I invented several stars and nuclear reactors&quot; and &quot;I invented a cheap and effective form of synthetic telepathy&quot;. It could also be meaningful to block you for block evasion (if it really is block evasion, but it seems very likely).
: An example of a page that was moved to user space as not good enough: [[User:TyEvSkyo/Particle Sphere Theory]], although it is pseudophysics, not pseudophilosophy. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 03:46, 9 April 2025 (UTC)
::Do you believe it's impossible to invent a nuclear reactor or envision a giant one capable of being a light and or electron source?
::Do you believe synthetic telepathy is fake or something? Feel free to search the patent office for &quot;synthetic telepathy&quot;. My invention does work, I have already applied it. and it produced extraordinary results.
::You claimed that nothing can or did exist at some time. Which to me is rank nonsense that you have not proven. Do you think your mere claims can rebut logical tautologies? [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 23:42, 9 April 2025 (UTC)
::: For the record, I requested a block at [[Wikiversity:Request custodian action#Block of MarsSterlingTurner]], which was implemented. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:26, 10 April 2025 (UTC)

A later note: by searching for &quot;nothing has the property of nothing&quot;, I found also the following 2025 discussion apparently started by MarsSterlingTurner (the user account there has a photo of the face): 
* [https://atheistdiscussion.org/forums/showthread.php?tid=9242 what pure logical tautologies can deduce], June 2025, atheistdiscussion.org
There are 5 pages of discussion. Apparently, the discussion members do show some engagement with this brand of low-grade material; they bother to respond. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:44, 28 August 2025 (UTC)

:The only argument you held against my work was that &quot;nothing can't imply nothing&quot;; but what do you think of the following;
:I know you will grant that nothing is nothing.
:But is can refer to the law of identity and therefore the biconditional; ↔
:Therefore you must grant that {}↔{}
:Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →
:Therefore you must grant {}→{}
:In other words ({}is{})→({}↔{})→({}→{}) [[User:AssumingNOTHING|AssumingNOTHING]] ([[User talk:AssumingNOTHING|discuss]] • [[Special:Contributions/AssumingNOTHING|contribs]]) 21:53, 29 August 2025 (UTC)
:: I tried to explain the matter at [[User talk:MarsSterlingTurner#Reflexivity of implication and empty set etc.]]. Further engagement with this kind of thinking/argumentation seems unlikely to be productive/helpful. I opened [[Wikiversity:Request custodian action#Block of AssumingNOTHING]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:18, 30 August 2025 (UTC)
:::This is a different argument; Did you grant the biconditional? [[Special:Contributions/2601:647:6512:3F4A:DD9D:BE18:F1E1:6F3A|2601:647:6512:3F4A:DD9D:BE18:F1E1:6F3A]] ([[User talk:2601:647:6512:3F4A:DD9D:BE18:F1E1:6F3A|discuss]]) 16:58, 30 August 2025 (UTC)
:::: No, I do not grant (or agree with) the biconditional &quot;{}↔{}&quot;, meaning &quot;the empty set if and only if empty set&quot;: the latter is nonsense, to me, and thus so is the former. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:27, 1 September 2025 (UTC)
:::::[[wikipedia:Law_of_identity|Law of identity - Wikipedia]]
:::::&quot;Modern logic
:::::In first-order logic, identity (or equality) is represented as a two-place predicate, or relation, =. Identity is a relation on individuals. It is not a relation between propositions, and is not concerned with the meaning of propositions, nor with equivocation. The law of identity can be expressed as , where x is a variable ranging over the domain of all individuals. In logic, there are various different ways identity can be handled. In first-order logic with identity, identity is treated as a logical constant and its axioms are part of the logic itself. Under this convention, the law of identity is a logical truth.&quot;
:::::If equality is a logical truth, then identity is a logical equality.
:::::Equals is logically equivalent to the biconditional. [[Special:Contributions/2601:647:6512:805F:98A3:A945:B2BD:C73A|2601:647:6512:805F:98A3:A945:B2BD:C73A]] ([[User talk:2601:647:6512:805F:98A3:A945:B2BD:C73A|discuss]]) 11:51, 1 September 2025 (UTC)
:::::: I don't know what the statement &quot;equality is a logical truth&quot; is supposed to mean or refer to. (Perhaps I should disengage; let us see.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:13, 1 September 2025 (UTC)
:::::::I mean &quot;law of equality&quot; is a logical truth as &quot;equality&quot; and &quot;identity&quot; are interchangeable and it says &quot;law of identity is a logical truth&quot; therefore &quot;law of equality is a logical truth&quot;, I said equality as a generalization for it's law.
:::::::Logical equality besides using = also can be expressed as the biconditional. Do you agree with that? [[Special:Contributions/2601:647:6512:805F:45BC:E93D:A300:BAE1|2601:647:6512:805F:45BC:E93D:A300:BAE1]] ([[User talk:2601:647:6512:805F:45BC:E93D:A300:BAE1|discuss]]) 19:35, 1 September 2025 (UTC)
:::::::: What reliable/serious sources use the term &quot;logical equality&quot; and what do they mean by it? Wikipedia is not a serious source, so I do not take [[Wikipedia: Logical equality]] seriously (it does not trace the term to a serious source, actually to any source). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:29, 2 September 2025 (UTC)
:::::::: As for &quot;identity is a logical equality&quot;: if ''logical equality'' is sameness of truth values (is it?), then this statement is false, since identity (the two-place relation) works not only for truth values but also for integers, cats, etc. (I am the author of [[An analysis of identity]].) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:37, 2 September 2025 (UTC)
:::::::: Consulting https://math.stackexchange.com/questions/3528250/what-is-the-difference-between-equality-and-logical-identity, this reinforces my request above for a definition/clarification of the term &quot;logical equality&quot;. I would also like to see seriously sourced definition of &quot;logical identity&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:47, 2 September 2025 (UTC)
:::::::::∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)
:::::::::∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={}) [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:23, 2 September 2025 (UTC)
::::::::::x↔y
::::::::::{}↔{} [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:25, 2 September 2025 (UTC)
:::::::::::[∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)]=(x↔y)
:::::::::::[∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={})]=({}↔{}) [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:27, 2 September 2025 (UTC)
::::::::::::[∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)]=(x↔y)
::::::::::::question; IF [∀x∀y[∀z((z∈x)↔(z∈y))] Then (x↔y) ?
::::::::::::x={}, y={}, z={}
::::::::::::therefore [(({}∈{})↔({}∈{}))↔({}={})]=({}↔{})
::::::::::::If (({}∈{})↔({}∈{}) then ({}↔{})
::::::::::::[[wikipedia:Axiom_of_extensionality#In_ZF_set_theory|Axiom of extensionality - Wikipedia]] [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:36, 2 September 2025 (UTC)
:::::::::::::Therefore (x=y)=(x↔y)
:::::::::::::Therefore ({}={})=({}↔{}) [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 19:39, 2 September 2025 (UTC)
: (Outdent) Anyone who has the first idea of the first-order logic (which is apparently employed here) must see that &quot;∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={})&quot; is sheer rubbish, even hilarious rubbish. The idea that you could take the meaningful ∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y) and get something meaningful from it by replacing all the variables with &quot;{}&quot; can hardly be more preposterous.
: This is perhaps not surprising given that this rubbish is comming from someone who, is his own words, has invented several stars (whatever that is supposed to mean). I am inclined to keep removing any more posts from the same source and keep on asking for an indef block for block evasion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:37, 3 September 2025 (UTC)
: Out of curiosity, I entered the first two lines from the above into Google Gemini 2.5 Flash:
::∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)
::∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={})
: I got the following response:
:: The second expression you wrote, ∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={}), is not a valid logical or mathematical statement. It appears to be a pattern based on the first expression, but the use of empty curly braces {} makes it meaningless. The first expression, however, is a fundamental axiom in set theory. It is a formal way of stating the Axiom of Extensionality. [...]
: Not too shabby. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:26, 3 September 2025 (UTC)
::I notice you deleted the post that corrected the issue you are claiming.
::Here is what I put in the same AI; [https://aistudio.google.com/prompts/new_chat?model=gemini-2.5-flash-image-preview Google AI Studio]
::∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)
::x={}, y={}, z={}
::therefore [({}∈{})↔({}∈{})]↔({}={})
::If (({}∈{})↔({}∈{}) then ({}↔{})
::therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]
::therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})
::therefore [({}={})→({}↔{})]
::Here was the response;
::5. [({}={})→({}↔{})]
::({}={}) is True.
::If ({}↔{}) is interpreted as True, then True → True is True.
::Summary:
::Your logical deductions seem to follow correctly, given the initial interpretation of {}∈{} as false and {}↔{} as true (representing self-equivalence or equality).
::Notice the AI accepts ({}↔{}) as true, due to my proof. [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 21:22, 3 September 2025 (UTC)
::: If you had the first idea of what you are doing, you would not need to correct your obvious nonsense (&quot;∀{}∀{}[∀{}(({}∈{})↔({}∈{}))]↔({}={})&quot;.
::: Let's play this game, though; perhaps it will be educational a bit despite the odds. I put your text above before &quot;Summary:&quot; to Gemini 2.5 Flash. The response I got is this: &quot;Your logical derivation contains several misunderstandings of both set theory and logic. Let's break down where the reasoning goes wrong. [...]&quot;.
::: However, I put &quot;Is the following accurate? ({}={})→({}↔{})&quot; to Gemini 2.5 Flash. As part of the response, it says 'The notation ({}={}) and ({}↔{}) does not have a defined meaning in formal logic.' I will note that '{}={}' does have a well-defined meaning in the first-order logic applied to the language of set theory, but &quot;{}↔{}&quot; does not. As a word of warning, Gemini gives slightly different answers when I repeat the query in new sessions; in this sense, it is not deterministic. As a further word of warning, this is an inquiry into the behavior of Gemini more than into the meaning of &quot;({}={})→({}↔{})&quot; since GenAI cannot generally be relied on/trusted.
::: Another interesting prompt is &quot;What does the following mean? ({}={})→({}↔{})&quot;. As part of a longer response, I get his:
::: 'This means:
:::: '&quot;If x is equal to y, then x has the same truth value as y.&quot;
::: 'This statement is always true in logic. If two variables or propositions are literally equal to each other, they must necessarily have the same properties, including their truth value.'
::: I don't agree with Gemini on this since x and y are not guaranteed to be truth values or objects carrying truth values. But things would improve thus: 'If x is equal to y, then x has the same truth value as y, as long as both x and y do have truth value.' However, this corrected sentence does not need to be true in Python or C++ since: &quot;equal&quot; would be interpreted as &quot;==&quot; rather than Python's is, and a class can redefine &quot;==&quot; to meaning something arbitrary and non-sensical as well as the bool casting operator, and then, in Python, possibly x == y while bool(x) != bool(y). If we try to carry this over to natural language, we may ask whether there is a significant difference between equality and identity; there is one in Python.
::: We see that Gemini contradicts itself: in response to the &quot;Is the following accurate? [...]&quot; prompt, it denies the expression has meaning, whereas in response to the &quot;What does the following mean? [...]&quot; prompt, it finds a meaning and explains it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:53, 4 September 2025 (UTC)
::::I put &quot;is the following true; ({}↔{})?&quot; in the AI
::::it said;
::::Yes, ({}↔{}) is true.
::::In propositional logic, &quot;p ↔ q&quot; (p if and only if q) is true if and only if p and q have the same truth value.
::::In this case, both sides of the biconditional are &quot;{}&quot;, which represents an empty set. When considering the truth value of an empty set, it is often interpreted as being vacuously true or having a consistent truth value in certain contexts (like in set theory or type theory, an empty type might be considered true if it implies nothing).
::::However, in the simplest interpretation for a basic logical statement like this, if we treat &quot;{}&quot; as a proposition, and both sides are identical, then they must have the same truth value. Therefore, the biconditional is true. [[Special:Contributions/&amp;#126;2025-50652-2|&amp;#126;2025-50652-2]] ([[User talk:&amp;#126;2025-50652-2|talk]]) 16:52, 4 September 2025 (UTC)
::::: I disagree with GenAI. When I try &quot;{}↔{}&quot; in Gemini, it interprets it as &quot;The empty set exists↔The empty set exists&quot;. That makes no sense to me; in the language of first-order logic, &quot;{}&quot; is not a truth-value bearing expression, and it does not mean &quot;The empty set exists&quot;. This is not going to get us anywhere.
::::: In your latest page [[User:205.154.222.227/Theory_of_monism_panpsychism]], you start with &quot;assuming nothing&quot; (except for logic, but you do not explicitly say so there) and arrive at the conclusion &quot;Therefore energy is consciousness&quot; and &quot;Therefore consciousness is eternal, omnipresent, omnipotent, energy&quot;. That is another overt nonsense: you need some axioms of ''energy'' and ''consciousness''; you cannot derive anything about them purely from logic axioms.
::::: Above, I tried to supply additional dose of patience, especially for the benefit of the reader. I am inclined to no longer respond since this is not going anywhere and the nonsensical character of you page should be obvious to almost anyone who has the first idea of logic and philosophy. The people at https://atheistdiscussion.org/forums/showthread.php?tid=9242 agree with me; after having engaged with you a bit, they blocked you (and so did we at the English Wikiversity). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:08, 4 September 2025 (UTC)
::::: For the record, I have removed a fairly long post by the user, consisting mostly of favorable assessments of his material by various GenAI. Such material is already posted by him elsewhere (under a different user); no need to duplicate it here. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:48, 4 September 2025 (UTC)

For ease of reference, some locations where the debated/discussed material is available in some form (helpful search terms: &quot;nothing has the property of nothing&quot;, &quot;something has always existed everywhere&quot;, &quot;something is self-causal&quot;):
* [[User:MarsSterlingTurner/Consciousness]], 2025
* [[User:MarsSterlingTurner/Ontology]], 2025
* [[User:MarsSterlingTurner/What there is Consciousness]], 2025
* [[User:MarsSterlingTurner/What there is Consciousness/Monism-Panpsychism-Pantheism Theorem]], 2025
* [[User:MarsSterlingTurner/What there is Consciousness/Monism Panpsychism Pantheism Theorem/MarsSterlingTurner]], 2025
* [[User:205.154.222.227/Theory of monism_panpsychism]], 2025
* [[User:HumbleBeauty/Proof of monism]], deleted in 2020
* [[Wikibooks:User:HumbleBeauty/Proof of monism peer review]], 2020
* [[Wikibooks:User:HumbleBeauty/Worship technology]], 2020
* [[Wikibooks:User:HumbleBeauty/Proof of monism peer review2]], 2021
* [[Wikibooks:User:HumbleBeauty/Proof of monism peer review3]], 2021
* [[Wikibooks:User:MeekFavor/proofs]], 2023
* https://www.quora.com/Is-it-possible-to-reach-a-universal-truth
* https://www.quora.com/Have-all-scientific-attempts-to-date-failed-to-explain-why-there-is-something-rather-than-nothing-by-starting-with-the-assumed-existence-of-something
* [https://atheistdiscussion.org/forums/showthread.php?tid=9242 what pure logical tautologies can deduce], June 2025, atheistdiscussion.org
* https://ctmucommunity.org/wiki/User:HumbleBeauty/proofs
* https://ctmucommunity.org/wiki/User:HumbleBeauty/Theory_of_monism_panpsychism
* https://ctmucommunity.org/wiki/Proofs_of_God
* https://rationalwiki.org/wiki/Debate:Logical_proof_of_the_existence_of_God, 2014‎, user Selfreasoning4all
* https://rationalwiki.org/wiki/Debate:Metalogical_reasoning_for_the_Divine, 2014‎, user Selfreasoning4all
* https://onlinephilosophyclub.com/forums/viewtopic.php?t=3590, 2010
* https://forum.atheistrepublic.com/t/is-this-proof-of-universal-consciousness/6893/4
* https://www.ilovephilosophy.com/t/monopantheism/21176?page=2, 2008, user MySiddhi with identifying photo; people trying to explain why he was wrong back in 2008
* https://www.stormfront.org/forum/t1269204/, 2019, bumped in 2025 (warning: stormfront.org is some kind of white supremacist and anisemitic web)
* https://www.stormfront.org/forum/t1425039/, 25 June 2025, user SubtleBeauty; user Lord Jim provides some impressively patient and competent responses
My guess is that any additional page with similar content should be outright deleted rather than moved to user space; there are more than enough user space page where IP anons can edit and expand this kind of material. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:33, 8 September 2025 (UTC)
: Expanded. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:39, 9 September 2025 (UTC)

== New Page Limit ==

I tried to respond on the talk page but my edit was blocked by the abuse filter (&quot;New User Exceeded New Page Limit&quot;). Could you please allow me to post my reply? [[User:Nazwa Shabrina|Nazwa Shabrina]] ([[User talk:Nazwa Shabrina|bicara]] • [[Special:Contributions/Nazwa Shabrina|kontrib]]) 12:20, 11 September 2025 (UTC)

:I want to fix the Sigma Spiral Constant page. [[User:Nazwa Shabrina|Nazwa Shabrina]] ([[User talk:Nazwa Shabrina|bicara]] • [[Special:Contributions/Nazwa Shabrina|kontrib]]) 12:22, 11 September 2025 (UTC)
:: For context, I tagged [[Sigma Spiral Constant]] for proposed deletion and placed a rationale/justification into the template.
:: You should have no difficulty editing the [[Sigma Spiral Constant]] page. As for editing [[Talk:Sigma Spiral Constant]], I would have thought you should be able to even as a new user. I just created [[Talk:Sigma Spiral Constant]] for the case that it would help you edit the talk  page. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:15, 12 September 2025 (UTC)
:::Now I can [[User:Nazwa Shabrina|Nazwa Shabrina]] ([[User talk:Nazwa Shabrina|bicara]] • [[Special:Contributions/Nazwa Shabrina|kontrib]]) 06:22, 12 September 2025 (UTC)

== Your proposed deletion of Basic Scratch Coding ==

Hello @[[User:Dan Polansky|Dan Polansky]],

I would like to inform you about your proposed deletion on the basic Scratch Coding page on Wikiversity. I am currently working on this project on Wikiversity and have created the project, and simply haven't found the time to contribute to the page, but I have managed to contribute to the page recently. I am currently improving the page, so I would like to remove the specific rationale.

Kind regards,

RailwayEnthusiast2025 —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 15:17, 15 September 2025 (UTC)

:P.S. : The Further Reading is on the 'Scratch' page, not on the 'basic Scratch Coding' page. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 15:18, 15 September 2025 (UTC)
:: A better venue for this topic is at [[Talk:Basic Scratch Coding]]. There I already provided some explanation. If that explanation is not enough or you have further questions, you can post there. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:20, 15 September 2025 (UTC)

== System- vs. User-Orientation ==

Why did you move the above from the main page to my user page. I am sick and tired.You may suppose it is my self-promotion. Then you are terribly.  It was my writing but ChatGPT's. No one could it write fairer than GPT.  Please geit it back. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 06:38, 25 September 2025 (UTC)
: 1) We are talking about [[User:KayYayPark/System- vs. User-Oriented Information Retrieval]]. There, we read &quot;Key contributions include: [...] K. Y. Park’s Direct Approach to Information Retrieval (1975), which emphasized user interaction with textual contexts (keywords and citations in context), representing an early user-oriented model.&quot; The page is not even marked as original research via a template and it does not state the author, thereby speaking in wiki voice. This is K.Y.Park (you) engaging in inappropriate promotion of a previously unpublished or poorly published thesis now available at [[A Direct Approach to Information Retrieval]].
: 2) As for &quot;No one could it write fairer than GPT&quot;, the idea that running something through ChatGPT (I am using Gemini) somehow consitutes a verification or validation (V&amp;V) is pretty absurd, except perhaps for a most superficial validation imaginable.
: As a result, I find my move fair and acceptable. My move is subject to review and reversal by full admins (or even semi-admins like myself?). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:44, 25 September 2025 (UTC)
: Even using Gemini does not validate your contribution, as per my new post here: [[User talk:KayYayPark/System- vs. User-Oriented Information Retrieval]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:47, 25 September 2025 (UTC)
: For the record: [[Wikiversity:Request_custodian_action#Block_of_User:KayYayPark]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:52, 25 September 2025 (UTC)

== Your opinion on Ninefold Resonance Theory ==

Dear Dan Polansky, I came across your interesting argument on [[Talk:May anyone call themselves a philosopher?]] and I was curious about your thoughts on my latest theory; the [[Ninefold Resonance Theory]]. If you would like to respond, please do so on [[Talk:Ninefold Resonance Theory|the discussion page]]! I look forward to hearing from you. Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 15:58, 28 September 2025 (UTC)
: I responded at [[Talk:Ninefold Resonance Theory]]. I fear you are going to be disappointed with my analysis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:16, 28 September 2025 (UTC)
::I didn't expect you to react with surprise and to see this as some kind of scientific breakthrough or anything, so I didn’t have high expectations! I really appreciate your feedback. As you can see, I have refined my theory a bit. I am curious to know if you think it's an improvement! [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 18:29, 28 September 2025 (UTC)

== Insertion of Photos ==

How can I insert private photos into the article? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 06:52, 29 September 2025 (UTC)
: You can upload photos (png?) on Commons (you have to be the copyright holder!). Home page: https://commons.wikimedia.org/wiki/Main_Page. Upload wizard: https://commons.wikimedia.org/wiki/Special:UploadWizard. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:53, 29 September 2025 (UTC)
::https://commons.wikimedia.org/wiki/File:B.C._Brookes_before_the_mailbox.jpg
::How can I  make the photo appear  from the above? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 07:37, 29 September 2025 (UTC)
::: Following the above link, I get &quot;No file by this name exists, but you can upload it.&quot; Your attempted upload must have failed? Other than that, the syntax &lt;nowiki&gt;[[File:B.C._Brookes_before_the_mailbox.jpg|thumb]]&lt;/nowiki&gt; would include the image on the page. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:39, 29 September 2025 (UTC)
::::Thank you so much. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 07:55, 29 September 2025 (UTC) 
::I've done it. Bother no more. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 07:54, 29 September 2025 (UTC)
::: You're welcome. I see, you uploaded this: [[:File:B.C. Brookes takes a motion to put a letter into the mailbox mouth.jpg]]. Although I think as a file, it is a bit too descriptive; it should ideally be shorter. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:00, 29 September 2025 (UTC)
::::Ya! I made it shorter. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 10:32, 29 September 2025 (UTC)
::::By the way, find my email adress from my &lt;nowiki&gt;[[User:KayYayPark]]&lt;/nowiki&gt; and email me anything so that I can know your email address. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 10:38, 29 September 2025 (UTC)
::::How stupid I am! My email address:  ishiakkum@google.com [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 12:12, 30 September 2025 (UTC)
::::: Your email address is a piece of sensitive, personally identifying information (it is one item that can, together with other items, help establish numerical identity of a human person). Revealing it on wiki is not necessarily what you want. Any bot can scan this page and then, someone can send you spam (even though you do have a spam filter, probably). You can ask your email address to be hidden (I think only a full admin can do it; I am a semi-admin/curator). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:16, 30 September 2025 (UTC)

== Peer review ==

I noticed with some alarm your combativeness with my deletion requets. On looking at your contributions, I see a substantial number of original research resources you have developed, few of which have been reviewed.

When are you going to seek peer review for them? How will you be arranging for such?

[[User:ජපස|ජපස]] ([[User talk:ජපස|discuss]] • [[Special:Contributions/ජපස|contribs]]) 16:11, 1 October 2025 (UTC)
: After I have seen your performance in another discussion in the English Wikiversity, I am inclined to disengage. I will respond to inquiries from Wikiversity custodians, for example. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:13, 1 October 2025 (UTC)
::Very well. I may start requesting many of your resources be deleted for lacking peer review. Apologies. [[User:ජපස|ජපස]] ([[User talk:ජපස|discuss]] • [[Special:Contributions/ජපස|contribs]]) 16:15, 1 October 2025 (UTC)
::: I am not aware of any policy-based requirement of a peer review for original research articles in the English Wikiversity. Nor has any custodian notified me of a requirement for peer review. Nor has, since the end of 2022, anyone notified me of such a requirement. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:17, 1 October 2025 (UTC)
::: Inspired by the above impulse, I asked myself where in the English Wikiversity one can find something like a peer review. I know of the following:
::: 1) [[Wikijournal]]s. They seem to take peer review quite seriously. That's something like a separate subproject of the English Wikiversity, so much so that there is a proposal to make it a completely standalone project.
::: 2) J.T.Neill is providing comments for his students in subpages of [[Motivation and emotion]]. However, it is not a ''peer review'', literaly speaking, since he is a teacher and not a peer/on the same level.
::: I cannot recall anything else. Perhaps there are more people like J.T.Neill; not that I remember right now.
::: I sometimes act as a partial reviewer by posting comments to pages, most recently here: [[Talk:Ninefold Resonance Theory]] (a page that is perhaps really for the user space rather than mainspace). But I assume no responsibility for comprehensiveness of these comments. (I have an extensive experience as a document reviewer from one of my previous jobs.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:52, 2 October 2025 (UTC)

== Clarification on authorship and licensing for Enhancing Web Browser Security through Cookie Encryption ==

Hello Dan,

I saw that you deleted and protected ''Enhancing Web Browser Security through Cookie Encryption'' citing possible copyvio from ResearchGate. I want to clarify that I am the original author of that ResearchGate publication. Since I hold the copyright, there is no violation. I am willing to release the text and figures under '''CC-BY-SA 4.0''', which is compatible with Wikiversity.

If needed, I can add an explicit license release statement on my userpage or the article’s talk page to confirm. The earlier move of the page was only due to a title mismatch (“Cookie Encryption”), not notability or copyvio.

Could you please advise how I should proceed to restore the page correctly with a clear license statement? I want to ensure the resource remains available under free licensing for Wikiversity readers.

Thank you for your time and guidance.

—Tom Joe James [[Special:Contributions/&amp;#126;2025-27242-55|&amp;#126;2025-27242-55]] ([[User talk:&amp;#126;2025-27242-55|talk]]) 15:56, 2 October 2025 (UTC)
: I checked https://www.researchgate.net/publication/391195563_Securing_and_Enhancing_Web_Browser_Security_through_Cookie_Encryption to see whether I can find any license statement. I find none. I cannot take a word from an anonymous IP account as reliable, in general; I have no means to verify the identity.
: But let's consider something else: what is the benefit from publishing the same text in the English Wikiversity? The text is already available as full text in researchgate.net, with references and properly formatted figures (which the page I deleted lacked). Surely the reader has all he needs in researchgate and suffers no loss from no article in the English Wikiversity? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:59, 2 October 2025 (UTC)
::The paper published in research gate is a pdf, which is not readable by LLMs or Web Crawlers properly.
::Publishing an Textual version of the paper ensures that the content is correctly indexed.
::And just like you, I've uploaded all the Wikiversity Rules to the GenAI and then asked for its opinions, it clearly stated that our paper is still eligible to be hosted in Wikiversity. (Source: ChatGPT &amp; Gemini after uploading Wikiversity several rule pages as Texts) [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 04:38, 3 October 2025 (UTC)
::: The link above is not a PDF (there is an additional button to get to a pdf). As for searchability by Google, I tried to search from  &quot;Securing browser cookies has been addressed by several&quot; (from the article) and it found researchgate and Wikisource. Meanwhile, the Wikisource page was deleted ([[S:Securing and Enhancing Web Browser Security through Cookie Encryption]]). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:44, 3 October 2025 (UTC)
::::Hello Dan, Kindly restore my previous comment that you have dismissed as &quot;GenAI&quot; Slop. So that others may follow our discussion. I have sent it so that you may understand my reasons. It was sent in a Good faith, not to spam or to vandalise.
::::The research gate link is is not pdf, but a document. Which cannot be viewed unless it is downloaded. There is a clear difference between a document and a web page. Even if the document gets indexed. It might not be read properly as it is structured differently. 
::::This is why a Wikiversity must be created for this document. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 04:54, 3 October 2025 (UTC)
::::: Anyone can see you post [[Special:Diff/2756895|here]]. I saw Wikipedia editors dismiss discussion input that is overt (and not signalled/indicated) GenAI slop. It seems a good idea to me to dismiss this kind of input. Sure enough, one can productively use GenAI to improve one's writing and discover items and ideas, but this overt voluminous slopping is a really bad idea, I think. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:57, 3 October 2025 (UTC)
::::::I used AI to expand my points to help you understand reasons behind my actions. Removing an entire reply from the main view instead of addressing it, is not a collaborative behaviour. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:16, 3 October 2025 (UTC)
::::: As for &quot;The research gate link is is not pdf, but a document. Which cannot be viewed unless it is downloaded.&quot;: That's not accurate: I viewed the document without downloading it (other than browser in a sense downloading any web page to serve it to the reader). The only problem with ResearchGate is that it seems to break down lines to achieve formatting so that Control + F on longer word sequences can fail. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:59, 3 October 2025 (UTC)
::::::Could you tell me how this paper is not eligible to be on Wikiversity?  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:18, 3 October 2025 (UTC)
::::::: Possible copyright violation, at a minimum. The April 2025 publishing venue (ResearchGate) has a page that does not indicate any license. Anyone responsible for the article would first have to contact ReserchGate and make them attach a license. Even then, we would need to clarify why ''double publication'' is an acceptable thing in this case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:24, 3 October 2025 (UTC)
::::::::I will add the license as soon as possible in the researchgate page.
::::::::And this is not a double publication, Research Gate is a self archive. And publishing in Wikiversity ensures accessibility. Improving visibility and indexing. We do not want our researchgate profile to be indexed. We want the content inside the paper to be indexed so that in the future LLMs can index it and train on the data. Research Gate acts as a formal preprint.
::::::::I hope everything is clarified.[[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:01, 3 October 2025 (UTC)
::::::::: Disclaimer: I am inexperienced with this matter. I make no promises on what will be allowed in Wikiversity. There are other admins (I am a semi-admin) around, and they can help resolve the relevant questions, I think. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:40, 3 October 2025 (UTC)
: Moreover, I asked Google Gemini: &quot;In Academic publishing, is it okay to publish the same article in multiple venues?&quot; It said: &quot;It is generally not okay to publish the same exact article in multiple academic venues. This practice is considered unethical and is known as duplicate publication or redundant publication.  [...]&quot;, with two links to sources. Sure enough, I went the cheap way and GenAI cannot be considered reliable, and would need to go to the sources and investigate the issue. But as a first quick impression, it reinforces my decision not to support recreation of the article in the English Wikiversity until someone can convincingly argue that GenAI is wrong. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:12, 2 October 2025 (UTC)
: For the record, I removed from this place a post that is a GenAI slop (see [[Special:Diff/2756895|this edit]] from [[User:Tomlovesfar]]). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:39, 3 October 2025 (UTC)
: (I am now responding here at the bottom, in part to a subthread above, to make it easier.)
: The principle implied in the proposed action is this:
:: &quot;The English Wikiversity should allow itself to be used as a double/duplicate repository for appropriately licensed content from ResearchGate, in part to drive LLMs.&quot;
: I am not sure/clear that this is a good principle, for the English Wikiversity. (Greetings to Kant, so hated by Rand.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 3 October 2025 (UTC)
::I’d like to clarify that this is '''not''' a case of double or duplicate publication in the academic sense.
::'''ResearchGate''' is a '''self-archiving platform''', similar to a preprint repository. It is not a peer-reviewed journal, and its content is not indexed or presented in a web-friendly, accessible format.
::'''Wikiversity''', on the other hand, is an open educational platform, with the goal of making knowledge freely reusable, remixable, and indexable (especially in plain-text formats). Publishing here is not a &quot;duplicate&quot;, but a '''different mode of dissemination''' serving different audiences.
::While driving LLM accessibility is a modern consideration, the '''primary goal is broader accessibility''' for human learners, researchers, and educators who may not access or index ResearchGate content easily.
::The version proposed for Wikiversity is '''freely licensed, educational, and structured for reuse'''
::As for Kant and Rand, I’ll just say I’m aiming for pragmatic clarity, not dogmatic idealism [[Special:Contributions/&amp;#126;2025-27507-81|&amp;#126;2025-27507-81]] ([[User talk:&amp;#126;2025-27507-81|talk]]) 07:09, 3 October 2025 (UTC)
::: That's it. I am not engaging with GenAI slop. I am keeping the above slop for the reader; I think I will remove any other slop. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:20, 3 October 2025 (UTC)
: As an item of public deliberation: we do not know whether the above interaction is with someone named &quot;Tom Joe James&quot;; neither self-disclosure by an anonymous IP editor nor relevantly named user account firmly establish the matter/the identity. It could well be just a disruption from an entity that for some reason wants to disrupt the English Wikiversity (why would they target such backwaters is not clear, but they could). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:02, 3 October 2025 (UTC)
::The IP addresses belongs to Tom Joe James. 
::I have been editing from my mobile phone, hence the guest edits.
::All of the above comments and replies were from me. You can check out my User:tomlovesfar and I have been contributing to the English Wikipedia under the same username tomlovesfar. 
::However, after updating my license on the research gate page, I would proceed to recreate the page under the same title but from my main account. That will be all from my side, Kindly do not delete any of the pages without proper discussion or warnings. 
::Best Regards, 
::Tom Joe James [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 08:15, 3 October 2025 (UTC)
::: Unless you send me a scan of your ID document, I have no way of reliably knowing who you are (but even ID documents can be fake). And as a curator/semi-admin, I guess I am not trusted to deal with this kind of sensitive information.
::: I guess you better chance is with other curators and custodians than with me. You can ask someone to review my approach. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:18, 3 October 2025 (UTC)
::::That isn't necessary, you can head over to my LinkedIn and under volunteering section I have mentioned my wikipedia contributions along with my username. 
::::or checkout my orcid id - 0009-0000-4896-9046
::::&lt;nowiki&gt;https://orcid.org/0009-0000-4896-9046&lt;/nowiki&gt;
::::That is more enough to prove it. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 08:25, 3 October 2025 (UTC)
::::: So I checked https://www.linkedin.com/in/tomjoejames-/; I do not see Wikipedia user account stated (&quot;Tomlovesfar&quot;)? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:30, 3 October 2025 (UTC)
::::::my apologies, I forgot that I had removed the screenshots. 
::::::I have updated it once again. Kindly check it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 08:35, 3 October 2025 (UTC)
::::::: There are screenshots, but a search for &quot;Tomlovesfar&quot; does not find anything ''in the text''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:36, 3 October 2025 (UTC)
::::::::Yes, I have updated it in the description.
::::::::&quot;User:tomlovesfar&quot; [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 08:39, 3 October 2025 (UTC)
::::::::: Okay, now we know that some entity has access both to the above IP account and to the LinkedIn account (really? am I jumping to conclusions?).
::::::::: I propose the following: 1) you make sure ResearchGate text has proper license there; 2) after that is completed, you contact some custodians (I am a curator) to review my approach. Before 1) is completed, any further communication is beside the point. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 3 October 2025 (UTC)
::::::::::Hello Dan,
::::::::::As requested, I have updated my ResearchGate publication with an explicit '''CC BY-SA 4.0 license statement'''. I’ve also mirrored this on my Wikiversity userpage for verification.
::::::::::Could you please reconsider restoring the article, or alternatively allow a custodian to review?
::::::::::Thank you,
::::::::::—Tom Joe James [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:14, 3 October 2025 (UTC)
: (Outdent) I skimmed the article at https://www.researchgate.net/publication/391195563_Securing_and_Enhancing_Web_Browser_Security_through_Cookie_Encryption. I also searched (Control+F) for &quot;Licen&quot;. I did not find anything. Where in the page should I look? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:16, 3 October 2025 (UTC)
::Securing and Enhancing Web Browser Security through Cookie Encryption
::* April 2025
::* DOI: 10.5281/zenodo.15287972License
::* CC BY-SA 4.0 you should find it under the Title
::[[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:18, 3 October 2025 (UTC)
::: I don't see the license item. I refreshed the browser as well, also hard refresh using Control+F5. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:20, 3 October 2025 (UTC)
::::[[File:Snapshot of a research gate article preprint.png|thumb]]
::::I have opened the link using incognito, It is still visible to me. 
::::Kindly Check this screenshot as well as this link. if possible send me your screenshot
::::https://www.researchgate.net/publication/391195563_Securing_and_Enhancing_Web_Browser_Security_through_Cookie_Encryption
::::You can find it under the date &amp; DOI number [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:25, 3 October 2025 (UTC)
::::: Still not seeing it; I tried a browser I hardly ever use, but I do not see it there either. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:27, 3 October 2025 (UTC)
::::: I also do not see the yellowish/orangish rectangle about ''preprints''. Perhaps you page is in ''preprint mode'' so I cannot see the change? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:39, 3 October 2025 (UTC)
::::::I think that you being served an old cache version. Maybe logging in would serve you the latest version. Or try clearing cache   [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:53, 3 October 2025 (UTC)
::::::https://zenodo.org/records/15287972
::::::Here is the Zenodo Link, Where our DOI is registered. The License is clearly mentioned under the tab &quot;Rights&quot;
::::::Hope this helps, Let me know how we can further proceed. [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 17:55, 3 October 2025 (UTC)
: (Outdent) I am still not seeing the license, and as I said, I used a different browser so the problem should not be in the browser cache.
: Moreover, I started [[Wikiversity:Colloquium#Publishing text from ResearchGate in Wikiversity as a copy]] to collect input from others. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:09, 3 October 2025 (UTC)
::I do not understand how this is possible. I have updated it in Zenodo, It should be visible to you.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 02:44, 4 October 2025 (UTC)

== Discussion regarding your behaviour as a curator ==

Hello, Just letting you know that I’ve raised a concern about our recent interaction at the [[Wikiversity:Colloquium#Concern regarding curator conduct User:Dan Polansky|Colloquium]] for review by other custodians. 

This is not meant to escalate conflict but to ensure transparency and get a fair, neutral perspective on how the discussion unfolded. [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 18:57, 6 October 2025 (UTC)

:I second that [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:03, 15 October 2025 (UTC)

== Preview ==

Do you use preview? It would be easier to look to the history, then if you use it a do not save with multiple small edits in a short period of time. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:11, 6 October 2025 (UTC)
: I follow an editing process that I find most productive for idea capture, which involves many rather small edits to my sandbox. I am investigating alternative options, but this is extremely productive. From my experience, editing in a long series of small edits is fairly common in the English Wikiversity, so I do not expect objections from native English speakers. As for the ease of reviewing the editing history, there is no need for anyone to inspect the editing history of my sandbox. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 20:15, 6 October 2025 (UTC)
::Your talk page is your sandobox? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:23, 6 October 2025 (UTC)
::: I thought you were referring to my editing in small edits of [[User:Dan Polansky/sandbox]], which is very extensive to the point that I am exploring alternatives (but it is very productive!) As far as I recall, my editing of my talk page does not suffer from this that much. It rather often happens that I save a response and only then realize I would like to amend it. So I do amend it. In the end, my posts are sometimes made in 2-3 edits. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 20:28, 6 October 2025 (UTC)
::::I see. Well, but that may lead to edit conflicts. I usually tend not to fix grammar erors send to the server not to cause such problems. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:36, 6 October 2025 (UTC)
::: As I was thinking about it, the above &quot;Your talk page is your sandobox?&quot; shows overt carelessness: 1) &quot;sandobox&quot; should not have &quot;o&quot;; 2) it should have a proper English question sentence form, &quot;Is your talk page your sandbox?&quot; But even that sounds sort of irritating to me; something is off. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:04, 8 October 2025 (UTC)
: (Outdent) Some questions to be clarified:
: 1) When editing one's talk page or another page where one participates on a discussion, should one carefully review the text before posting it, to avoid edit conflicts?
: 2) When (in relation to the above), even with best care, one notices a need to edit one's post to a discussion, and when no one has responded yet, should one avoid further editing one's post, in view that this may lead to an edit conflict, causing a complication for the responding person?
: 3) If the answer to item 2) is no, one should take risk of edit conflict: should the person suspecting and fearing an edit conflict edit their response e.g. in Notepad and avoid posting a response until some reasonable time elapses, e.g. 5 minutes?
: 4) When one is not a native speaker of the language of communication (here English), should one remind oneself that one is composing in a foreign language, which will naturally lead to more mistakes that need correcting? Should one therefore be tolerant/kind toward oneself, and accept the repeated edits as a cost of trying to deliver value in a foreign language?
: --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:49, 8 October 2025 (UTC)

== Seperate See Also Page  ==

Why can the seperate '''[[See Also]]''' page not be transplanted into another page with &quot;{{See_Also}}&quot;? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 08:05, 10 October 2025 (UTC)
: Instead of responding to the above, let me point out I recently opened a request: [[Wikiversity:Request custodian action#Block of User:KayYayPark or prohibition of new pages]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:13, 10 October 2025 (UTC)
::Why do you like me to be blocked? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 15:01, 10 October 2025 (UTC)
: Let me note that it is quite remarkable that you apparently forgot the skills that you had under [[User:KYPark]] account. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:52, 10 October 2025 (UTC)
::Yes, I am supposed to have lost a lot. What a pity! It seems so because of my old age. The above question is quite embarassing, because I confirmed it from my old page. There is nothing wrong with my present practice. Anyway you must know what's wrong with &quot;See Also&quot; of [[Memex]]. Please tell me. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 15:08, 10 October 2025 (UTC)
::There, &lt;nowiki&gt;{{See_Also}}&lt;/nowiki&gt; returns Template:See Also in red color. Why not returning properly? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 15:16, 10 October 2025 (UTC)

== Regarding Wikiversity:Colloquium ==

Hello Dan,
After looking at https://en.wikiversity.org/wiki/Wikiversity:Colloquium#Concern_regarding_curator_conduct_User:Dan_Polansky and my own personal experiences, I would like to politely ask you to instead of asking for deletion of my learning resources, try and help me improve and give advice. After all, I am fairly inexperienced at Wikiversity. I understand that you might be helping at Wikiversity, but for similar situations with this deletion matter, please hand it over to other curators or custodians such as @[[User:Atcovi|Atcovi]] or @[[User:Koavf|Koavf]]. In the meantime, please can you give me advice on how to improve, as I have seen you are fairly experienced here.

Thanks, 

RE —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 20:16, 13 October 2025 (UTC)

:As an immediate concern, if there is some learning resource that in principle belongs here but in practice is insufficient or inappropriate, userfying it is a good short-term solution. Just move a page from ''foo'' to ''User:RailwayEnthusiast2025/foo'' and build it up until it's acceptable and move it into the main namespace. I'm not knowledgeable enough about any particular resource or making any implicit claim about the quality of your work, just saying that this is a perfectly amenable solution. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:26, 13 October 2025 (UTC)
:: @RailwayEnthusiast2025: In [[Special:Diff/2745947|this edit]] from 18 September 2025, you removed request for deletion from [[Basic Scratch Coding]], which was really a request for ''moving to user space''. Rather than ''delete'' pages, I usually move them to user space, based on Wikiversity tradition; an exception is when there is a suspicion of copyright violation, but even in such cases, I plan to use [[Template:Copyrighted]] much more, in case of doubt, instead of immediate deletion.
:: As it is, [[Basic Scratch Coding]] is not ready for mainspace. Ideally, it should be moved to user space, after which you will have ''indefinite time'' to work on it.
:: A relevant discussion already took place at [[Talk:Basic Scratch Coding]], where I said: &quot;I now added a proposed deletion since in this state, the course is of little use. There is a protective period of 3 months before deletion, expiring on December 14, 2025. Even then, the page would typically be not outright deleted but rather moved to user space. I feel this is not too harsh; by Dec 14 2025, over a year will have elapsed from this page having been started&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 21:02, 13 October 2025 (UTC)
:: A note, perhaps an aside: I got involved with pages by Harold Foppele after he pinged me at [[Talk:Quantum A Matter Of Size]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:27, 13 October 2025 (UTC)
:::What have i to do with all this? DFX [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:10, 15 October 2025 (UTC)
:::Actually, you are correct. If a project is in mainspace, it should help participants learn. Draft namespace and userspace is a place in where you can develop your pages. Thanks for helping me understand. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:50, 6 November 2025 (UTC)
:::: Thank you. Let me add that I am not the boss here: if I assess a page as not fit for the mainspace, you or anyone else can challenge my decision at [[WV:RFD]] as a request for undeletion (moving out of mainspace is quasi-deletion) and then we call collect input from others on your artifact (page or a set of pages). I am fallible and make mistakes, more than I care to admit, but I am trying to learn from them, again and again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 19:11, 6 November 2025 (UTC)
::So creation of a learning materials is no longer community driven? Why they cannot be developed in the main name space? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:35, 16 October 2025 (UTC)
:::I don't know. I only really moved these pages to my user space due to @[[User:Dan Polansky|Dan Polansky]] giving the advice that my projects weren't good enough for &lt;s&gt;my namespace&lt;/s&gt; mainspace. &lt;s&gt;It &lt;/s&gt; His language seemed as if he were to drive me off Wikiversity and as if my contributions weren't valued. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 16:33, 16 October 2025 (UTC)
::::*mainspace —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 16:34, 16 October 2025 (UTC)
::::My question was for Justin, as you can see I am idnenting it below his post. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:02, 16 October 2025 (UTC)

== Is it you? ==

Is it you who has deleted [[Korean/문맥색인요약]], and some other Korean articles under [[Korean]]? Then please explain why these are unfair to be there under [[Korean]]? -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 13:29, 16 October 2025 (UTC)

:You can [https://en.wikiversity.org/wiki/Korean/%EB%AC%B8%EB%A7%A5%EC%83%89%EC%9D%B8%EC%9A%94%EC%95%BD?action=edit&amp;veswitched=1 see here] @[[User:KayYayPark|KayYayPark]], who have deleted that page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:49, 16 October 2025 (UTC)

:: Thank you but I am sorry the above link links to the deleted page. -- [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 08:26, 17 October 2025 (UTC)

: The page was deleted by [[User:MathXplore|MathXplore]] with the summary &quot;Not English. See https://www.wikiversity.org to find where contributions may be appreciated&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:11, 3 November 2025 (UTC)

== Questions about qualifications and questioning expertise ==

Do you think it is appropriate to [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&amp;oldid=2760143#Your_qualification ask users what their qualifications are] and [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&amp;oldid=2760190#Contest_removal_of_article question their qualifications]? Not me. Wikiversity is not some kind of professional workplace where we would copy the structures of a real university and decide who has what degrees or expertise. Wikiversity is an open educational space that allows even those who are not qualified to teach and learn. The question is also how you can recognize the qualification in the field of physics when you yourself write that [[User:Dan Polansky|you are a programmer]]. But even if you could, there are certainly more appropriate ways dispute the quality of the resource, as the discusion on the resource talk page or [[:Template:Citation needed]]. If you are unhappy with these traditional or available ways you may propose new policy on such a content. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 16 October 2025 (UTC)
: The policy proposal [[Wikiversity:Artificial intelligence]] states:
:: &quot;Generative artificial intelligence large language models may be able to generate text that may be suitable for inclusion on Wikiversity. However, to be acceptable:
::: &quot;The contributor should be an expert on the topic
::: &quot;[...]&quot;
: I suspected that the user used GenAI. I therefore found it appropriate to investigate whether he was an expert on the topic of quantum mechanics, or at least had a half-decent basic knowledge. Out of courtesy, I expressly indicated that a response to my inquiry was not required by a policy (since this is a policy ''proposal'', not a policy): &quot;Since you are entering articles on advanced physics, it would be worthwhile to know your qualification, e.g. university-level degree. Disclaimer: as far as I know, Wikiversity has not introduced any codified requirement of your disclosing such qualification.&quot;
: If the English Wikiversity determines that such an inquiry is undesirable/inappropriate, I can accept that, even though somewhat begrudgingly, since I think a ''voluntary'' disclosure of one's qualification is a very good thing in the context of the English Wikiversity (and should even be ''mandatory'' for those who insert GenAI output?).
: On a relational note, I do not think you are in the position to make edicts telling me what I may and may not do; you can make proposals at best. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:00, 3 November 2025 (UTC)
::It has been [[Wikiversity:Etiquette|determined here]] since 2006. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 17 November 2025 (UTC)
::: The page [[Wikiversity:Etiquette]] has neither policy nor guideline status. The consensual support for that page is unclear. Whether anyone carefully reviewed the page with the intent of finding errors and possible harm resulting from the page is unclear. For instance, the page says &quot;Use edit summaries&quot;, but that is very often ignored in the English Wikiversity; this suggests that the page as a whole does not have consensus. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:40, 18 November 2025 (UTC)

== Testing newbies is not civil ==

I am sorry Dan. You are not here in the position of an authority [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&amp;oldid=2759624 to test newbies]. You can dispute content - if you see copyright violation you can ask for the prove, that the author uses its text or delete it. But why you investigate the users this way? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:01, 16 October 2025 (UTC)
: My response above at [[#Questions about qualifications and questioning expertise]] is applicable here. By asking &quot;Can you, in your own words, quickly describe the double slit experiment here?&quot;, I tried to determine whether the user had at least a very basic idea of quantum mechanics, in part since that was required by the policy proposal, and in part since it has bearing on ''authorship'' of content being entered into Wikiversity.
: The link you provided, https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&amp;oldid=2759624, shows the editor to be very problematic, yet you take his side and take no reservation to his conduct. (Part of the response self-censored as perhaps better avoided, after all.) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:45, 3 November 2025 (UTC)

== [[Korean/게르만어와한국어]] ==

Is it you who deleted the above page? --[[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 11:20, 22 October 2025 (UTC)

This was edited directly so that I have no copy. I was going to upload it to x.com. Please do me a favor to return that deleted copy back to me either way, say, via email ishiakkum@gmail.com.  [[User:KayYayPark|KayYayPark]] ([[User talk:KayYayPark|discuss]] • [[Special:Contributions/KayYayPark|contribs]]) 11:46, 22 October 2025 (UTC)
: Email sent. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:38, 22 October 2025 (UTC)
: The page was deleted by [[User:MathXplore|MathXplore]] with the deletion summary &quot;Not English. See https://www.wikiversity.org to find where contributions may be appreciated&quot;. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:05, 3 November 2025 (UTC)

== MediaWiki questions and answers ==

Moving the issue to the [[Wikiversity:Colloquium#MediaWiki questions and answers|Coloquium]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:17, 26 October 2025 (UTC)
: The above is in reference to [[User talk:Juandev#Page move]], September 2025, where I spent quite a bit of effort to explain why keeping [[User:Juandev/MediaWiki questions and answers]] in mainspace is a bad idea.
: You can request undeletion of pages (move to user space is quasi-deletion) in [[WV:Requests for deletion]]; that is the standard process. On the other hand, using Colloquium is perhaps okay, especially if you want to clarify ''general principles''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:08, 3 November 2025 (UTC)

== Community Review ==

Please see [[Wikiversity:Community Review/Dan Polansky]]. I have temporarily removed Curator rights pending the outcome of the discussion. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 06:22, 16 November 2025 (UTC)
: @[[User:Mu301]]: I responded there. Can you clarify why you removed the rights before the discussion rather than after the discussion? Did I misuse the rights in any way, or was there a risk of my misuse of the rights? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:55, 16 November 2025 (UTC)

== Night mode unaware.. ==

Hitting a limit as to what I can usefully patch at my access level. 

I've sandboxed a new version of Collapse top. ([[Template:Collapse top/sandbox]] Please review. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:22, 17 November 2025 (UTC)

: The remaining non content or userspace instances of the Lint concerned are mostly template transclusions, which need the protected templates to be patched :(. I can't do this.  so would appreciate someone with the relevant access doing so, based on the approach I have been taking. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:22, 17 November 2025 (UTC)
:: I am not clear what I can do for you; I was a curator, and currently, I do not have the curator rights (see also [[Wikiversity:Community Review/Dan Polansky]]). So either I can do that for you if and after I get the curator flag back, or perhaps you are better off going to Colloquium. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:25, 17 November 2025 (UTC)

== Trolls ==

Do you know that it's not appropriate to publicly call someone a troll? It only leads to an escalation of the conflict. If someone bothers you, it doesn't mean that they are a troll. It's just a personal incompatibility. In order for someone to be called a troll, there must be a consensus of several users, and the discussion about it often takes place offline to avoid developing a conflict. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:50, 17 November 2025 (UTC)
: I am not clear that engaging the above multi-wiki convicted troll (intentional disruptor causing significant harm/economic loss) Juandev is a good idea (more at [[Wikiversity:Community Review/Dan Polansky]]). For anyone reading this and wanting to know about the topic of troll-labeling, I will write something.
: My understanding is quite different. The above makes it seem as if there is some universally accepted and universally valid principle unconditionally prohibiting troll-labeling. But I have seen a different practice. I have myself said of user AP295 that I wondered whether his behavior was a kind of trolling and no one complained ([[User talk:AP295]], &quot;I wonder whether what I am seeing is a certain kind of trolling.&quot;). AP295 was indeffed in the English Wiktionary with reference to trolling, and the block of AP295 at Meta also referenced trolling, per https://meta.wikimedia.org/w/index.php?title=Special:CentralAuth&amp;target=AP295. Disregarding now the observable actual practice, early detection and neutralization of intentionally disruptive editors is vital for the health of the project, as the English Wikiversity should know by now only too well, with memory of major disruptors Abd and Marshallsumter. One may perhaps wonder whether the label &quot;disruptor&quot; is better than &quot;troll&quot; since it does not impute intent, but in the case of Harold Foppele, the disruption being intentional is manifest. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:34, 18 November 2025 (UTC)

== 1-week block ==

See [https://en.wikiversity.org/w/index.php?title=Wikiversity:Community_Review/Dan_Polansky&amp;diff=prev&amp;oldid=2775054 here] for the reasons. Inappropriate usage of your talk page will also result in you losing access to it. 

Also, your [[User:Dan Polansky/User Juandev|page on Juandev]] is not welcomed here, and per your comment: &quot;If some custodian considers this manner of collection inappropriate, I can accept that (even if begrudgingly) and I request that the content of the page is sent to my email if the page is deleted&quot;, I will send the content of this page to your email so you can have it for your ''own'' records and have it be off-wiki. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:09, 21 November 2025 (UTC)
: I will copy your reason here for ease of reference, from [[Special:Diff/2775054]]: &quot;I have blocked Dan for 1 week due to the obvious disruptive conduct taking place in this community review. Dan is using this community review as a way to launch unsupported attacks against other users, including [[User_talk:Atcovi#Child_psychology/Summary_of_child_psychology_(cheat-sheet)|allegations of me using &quot;GenAI&quot; to create my pages]] and an [https://en.wikiversity.org/w/index.php?title=Wikiversity:Community_Review/Dan_Polansky&amp;diff=prev&amp;oldid=2775006 unnecessary comment on another user's contributions to our project]. Also, the [[WV:Canvassing|canvassing]] taking place here is also indicative of Dan's lack of remorse for his actions. Lastly, the LTA, who is also asserting Dan's racist comments (which Dan has not addressed nor expressed remorse for), leaves a bad impression on our community. I want to remind everyone that [https://meta.wikimedia.org/wiki/Requests_for_comment/Shut_down_Wikiversity proposals for this project's removal have took place in the past], and we surely don't want another discussion to be brought up for WV's removal. I'm hoping that the block imposed here will give some time for the community to productively discuss this community review, and for Mike to make his comments or give his final verdict. If Dan returns after the block and continues to pursue his disruptive, slanderous campaign, a harsher block will be imposed. Thanks.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:25, 21 November 2025 (UTC)
: Can you clarify which canvassing are you talking about? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:25, 21 November 2025 (UTC)
: Can you explain why you deleted [[User:Dan Polansky/User Juandev]]? The page contained detailed evidence supporting a measure agains Juandev. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:26, 21 November 2025 (UTC)
: As for your statements that I am making allegations, I have ''asked a question'', not alleged: &quot;Since the pages have at least a modicum of GenAI wibe, I would like to ask you whether you used GenAI to compose these pages, whether in whole or in part. The policy draft/policy proposal concerning GenAI is WV:Artificial intelligence.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:33, 21 November 2025 (UTC)
: As for &quot;Dan's racist comments&quot;: my task as the wikidetate/pro-con-analysis creator was to provide arguments for both sides rather than only arguments I agree with. I therefore provided the most salients arguments I could figure out that were in the air, as it were. These were not ''my'' comments; these were comments of an abstract potential Wikidebate participant, something like a literary figure. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:39, 21 November 2025 (UTC)
:: As per [[Wikidebate]], &quot;Wikidebate is a collaborative learning project to gather and organize ''all'' arguments on controversial issues&quot;, italics mine. Granted, the word &quot;all&quot; needs to be interpreted somewhat non-literarily; the arguments need to have a modicum of relevance and strength. But it is not suggestive of censorship, not even self-censorship. You did not yet take the opportunity to open a Colloquium discussion about [[Wikidebate]] and the debate format; why not? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:10, 21 November 2025 (UTC)
: Okay, I noticed [[Special:Diff/2775076|your another edit]] where you seem to indicate that the following is a &quot;canvassing&quot;. Is that right, is that canvassing? &quot;As an aside, can you help support my proposal at [[Wikiversity:Colloquium#General ban on direct use of GenAI output with exceptions]] so that we can curb/limit a developing harm of the English Wikiversity, the mounting volume of mainspace pages with predominantly GenAI-generated output?&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:47, 21 November 2025 (UTC)
: I would like to remind you that the blocking tools include the option to block specifically [[Wikiversity:Community Review/Dan Polansky]], if that is the main part of rationale. If you change the block to that page specifically, or also to mainspace, I will be able to continue post my analyses to my user space, for the benefit of myself and perhaps others. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:32, 21 November 2025 (UTC)

== Three-month block ==
I was blocked by user Atcovi for the following: &quot;continued disruptive editing, unwillingness to cooperate, cross-wiki harassment, and further time-wasting&quot;.

I do not know what that refers to. No diffs or links were provided. I consider the unsubstatiated statements and the block itself highly inappropriate. I find it libelous. It is not acceptable to talk in generalities like that without a shred of substantiation, to my mind.

The overt disruptor Harold Foppele continues unabated, with no block against him at all. This is all just absurd.

Today, I opened the following discussion, in which Atcovi should provide answers:
* [[Wikiversity:Colloquium#PSYC322 - Adolescent Psychology and potential plagiarism]]
The discussion is about pages by Atcovi and his ''potential'' plagiarism. His use of block tool against me after this is, to my mind, highly inappropriate.
What has become of the English Wikiversity? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:57, 17 December 2025 (UTC)

== Request to Jtneill ==
@[[User:Jtneill]]: you were my mentor when I was a curator. Let me ask: is the indefinite block of me appropriate? Why is not a one-year-block more than enough? Did I really ''disrupt'' Wikiversity or did I rather do a lot to help it (keyword: Marshallsumter)? Are there any uses of my curator tools that were determined to be inappropriate? Thank you and sorry to bother you. I am quite distressed.

(I also requested this back in November 2025: [[Wikiversity:Request custodian action/Archive/29#Indefinite block for Harold Foppele]]. Maybe I was too clumsy or inexpert in that request; maybe the substantiation was not good enough. The user account is now globally locked. The cleanup after this disruptive account in the English Wikiversity is still pending, I think; by my assessment, all pages he created should be moved out of mainspace to his user space.

It was also in November when I requested this: [[Wikiversity:Colloquium/archives/November 2025#General ban on direct use of GenAI output with exceptions]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:30, 10 March 2026 (UTC)

@[[User:Prototyperspective]]: You just pinged me, but I am indefinitely blocked since today. I am not going to respond on substance since that could be seen as a block evasion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:37, 10 March 2026 (UTC)

On the GenAI note, let me note that adopting [[Wikiversity:Artificial intelligence]] as is is ''super unwise''; it is a de facto go head to placing GenAI output in mainspace in volume. While there is &quot;The text should ideally be revised and rewritten by the contributor&quot;, the word &quot;ideally&quot; makes it optional. There is now a risk that this bad proposal is going to be adopted, as per [[Wikiversity:Colloquium#Wikiversity:Artificial intelligence to become an official policy]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:53, 10 March 2026 (UTC)

:Consensus has been reached to block you indefinitely per the Community Review. Your continued lack of remorse and abuse of your talk page privileges has led you to this point. I earnestly suggest you take this time to reflect on your actions and way of handling high-stake situations, and I hope you return to our project in the near future with these issues recitified after a successful appeal. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 23:10, 10 March 2026 (UTC)

:{{ping|Dan Polansky}} Sorry, I haven't been around much over last few months. It seems that your strong views and associated actions, whilst well intended, caused too much friction and led to consensus to block. Perhaps there are some lessons here that, if learnt, could lead to undoing the indefinite block. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:03, 24 March 2026 (UTC)</text>
      <sha1>5dyjft0hzx7fwco7nir61ceis149qxe</sha1>
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  </page>
  <page>
    <title>User:Alandmanson</title>
    <ns>2</ns>
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      <id>2819130</id>
      <parentid>2819070</parentid>
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      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Project: African Arthropods */</comment>
      <origin>2819130</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3676" sha1="i35owlphrg26cwkydjf0x44ifvn5obl" xml:space="preserve">&lt;!--Info--&gt;
https://www.archive.org
{{list subpages|Alandmanson|User}}

===[[African Arthropods|Project: African Arthropods]]===

;[[African Arthropods/Chelicerates|African Chelicerates]]
:Arachnids and sea spiders — No sub-pages yet.
;[[African Arthropods/Crustaceans|African Crustaceans]]
:Including branchiopods, barnacles, crabs, lobsters, crayfish, shrimp, fish lice, tongue worms, and ostracods — No sub-pages yet.
;[[African Arthropods/Hexapods|African Hexapods]]
:[[African Arthropods/Insects|African Insects]]
:* '''[[African Arthropods/Diptera|Diptera]]''' 
:**[[African Arthropods/Acalyptrate flies|Acalyptrate flies]]

:* '''[[African Arthropods/Hymenoptera|Hymenoptera]]'''
:**[[African Arthropods/Chalcidoidea|African Chalcidoidea]]
:***[[African Arthropods/Eulophidae|African Eulophidae]]
:***[[African Arthropods/Encyrtidae|African Encyrtidae]]
:***[[African Arthropods/Afrotropical Encyrtidae Key|Key to the genera of Afrotropical Encyrtidae]]
:***[[African Arthropods/Chalcid wasps with branched antennae|African chalcid wasps with branched antennae]]
:***[[African Arthropods/Wasps associated with plant galls|Wasps associated with plant galls]]
:**[[African Arthropods/Diaprioidea|African Diaprioidea]]
:**[[African Arthropods/Platygastroidea|African Platygastroidea]]
:**[[African Arthropods/Aculeata|African Aculeata]]
:***[[African Arthropods/Apoidea|African Apoidea]]
:****[[African Arthropods/Ampulicidae|African Ampulicidae]]
:****[[African Arthropods/Crabroninae|African Crabronidae]]
:****[[African Arthropods/Sphecidae|African Sphecidae]]
:****[[African Arthropods/Pemphredonidae|African Pemphredonidae]]
:****[[African Arthropods/Philanthidae|African Philanthidae]]
:*****[[African Arthropods/Philanthus|South African species of Philanthus]]
:***[[African Arthropods/Eumeninae|African potter wasps]]

:* '''[[African Arthropods/Lepidoptera|Lepidoptera]]'''

;[[African Arthropods/Myriapods|African Myriapods]]
:Centipedes, Millipedes, Pauropodans, Symphylans — No sub-pages yet.&lt;br&gt;&lt;br&gt;

;Arthropods in South Africa
:[[African Arthropods/Ferncliffe Nature Reserve|Ferncliffe Nature Reserve]]
:[[African Arthropods/Arthropods on ''Ficus burkei''|Arthropods on ''Ficus burkei'']]
:[[African Arthropods/Hymenoptera of South Africa|Hymenoptera of South Africa]]
:[[African Arthropods/Pompilidae of South Africa|Pompilidae of South Africa]]
::[[African Arthropods/Pompilidae of SA with yellow wings tipped black|Pompilidae of SA with yellow wings, wingtips black]]
::[[African Arthropods/Pompilidae of SA with dark, blackish wings|Pompilidae of South Africa with dark, blackish wings]]

&lt;br&gt;

===To Do===
Working on:
[[User:Alandmanson/Hymenoptera of Africa]]

Microgastrine cocoons in a net: &lt;br&gt;
* http://www.waspweb.org/Chalcidoidea/Eupelmidae/Eupelminae/Eupelmus/Eupelmus/Eupelmus_species_2.htm
* https://www.waspweb.org/Ichneumonoidea/Braconidae/Microgastrinae/Glyptapanteles/Glyptapanteles_acraeae.htm
* https://commons.wikimedia.org/wiki/File:Microgastrinae_cocooncocoon_iNat_42943906.jpg
* https://www.inaturalist.org/observations/38150348
* https://www.inaturalist.org/observations/144355729
* https://www.inaturalist.org/observations/39807090
* https://www.inaturalist.org/observations/145817446&lt;br&gt;


[[Crop_production_in_KwaZulu-Natal|Project: Crop_production_in_KwaZulu-Natal]]

[[Crop production in KwaZulu-Natal Annotated Bibliography]]

[[Information for smallholders in KwaZulu-Natal]]

[[Crop_production_in_KwaZulu-Natal/Climate-smart_Agriculture|Climate-smart Agriculture in KZN]]

[[Plant propagation]]&lt;br&gt;
&lt;br&gt;


[[Animal Phyla/Arthropoda]]&lt;br&gt;

[[:Category:Animals]]&lt;br&gt;

[[:Category:Zoology]]&lt;br&gt;

[[:Category:Entomology]]</text>
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  <page>
    <title>User:Alandmanson/sandbox</title>
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      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>liris</comment>
      <origin>2819241</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19636" sha1="ruh0yof32s5ebt687tpclpf6ewv1alr" xml:space="preserve">&lt;!--Info--&gt;
== Rough guide to Liris species of South Africa ==
=== Wings dark fuscous ===
Very dark wings, often with violet reflections
&lt;gallery mode=packed heights=200&gt;
Liris inaturalist 197329080.jpg

&lt;/gallery&gt;
=== Wings hyaline ===
Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin
&lt;gallery mode=packed heights=200&gt;


&lt;/gallery&gt;
=== Wings flavo-hyaline, often with a fusous (darker) apical margin ===
*Liris haemorrhoidalis*, *Liris bembesianus*, *Liris solstitialis*
&lt;gallery mode=packed heights=200&gt;
Liris on Crassula iN 42678436 03.jpg

&lt;/gallery&gt;

= Pompilidae of South Africa =
== South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips ==
&lt;gallery mode=packed heights=200&gt;
Inaturalist 258649905 b.jpg
Hemipepsis hilaris - inaturalist 10850475.jpg
Cyphononyx decipiens inat 26259647 b.jpg
Tachypompilus ignitus inaturalist 311015843 02.jpg
Pompilidae 2021 12 12 inaturalist 313386858 04.jpg
Pompilidae 2020 04 13 inaturalist 43563902 06.jpg
&lt;/gallery&gt;
*The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent.
*
&lt;br&gt;

== South African Pompilidae with fore-wings fuscous (black or very dark) ==
*The wings often have green-blue-violet reflections.
&lt;gallery mode=packed heights=200&gt;
Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus''
Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus''
Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus''
Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex''
Pompilidae_2019_05_28_0256.jpg|
Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp.
&lt;/gallery&gt;

&lt;br&gt;
=== Species  with black antennae, legs, head, thorax and abdomen ===
Some parts may be brown.

*''Java atropos''
*''Cyphononyx obscurus''
*''Hemipepsis vindex''
*''Hemipepsis vespertilio''
*''Hemipepsis braunsi''
*''Batozonellus fuliginosus''
&lt;br&gt;
=== Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red ===
*''Cyphononyx optimus''
*''Paracyphononyx zonatus''
&lt;br&gt;
&lt;br&gt;

== South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline ==
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg
Pompilidae inaturalist 46961473.jpg
Pompilidae iN 144781033 03.jpg

&lt;/gallery&gt;
*With fuscous (darker) wing apex
*One or two fuscous bands (faciated or bifaciated)
*Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted)


&lt;br&gt;
== South African Ceropalinae ==
Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020]
*''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia)
*''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region)
*''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa)
*''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe)
*''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa)
*''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?)
*''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) 
*''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe)
*''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda)
*''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa)
**''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe)
**''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa)
*''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa)
**= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319
*''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania)
*''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe)
&lt;br&gt;

==Afrotropical Ceropalinae ==

Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).&lt;ref name=Madl2020&gt;Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84.
[https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]&lt;/ref&gt;


Ceropalinae can be defined by:&lt;ref name=Brothers1993&gt;Brothers, D. J. &amp; Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. &amp; Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143&lt;/ref&gt;&lt;ref name=Waichert2015&gt; Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., &amp; Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]&lt;/ref&gt;

== Genera and species of Afrotropical Ceropalinae ==

This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa &amp; Schmid-Egger, 2025)&lt;ref name=CoL2025&gt; Kroupa, A. S., &amp; Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv &lt;/ref&gt; and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].&lt;br&gt;

=== Genus ''Ceropales'' ===
*''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia)
*''Ceropales angolaensis'' Móczar, 1989 (Angola)
*''Ceropales arnoldi'' Móczar, 1988 (Namibia)
*''Ceropales atra'' Móczar, 1991 (Botswana)
*''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region)
*''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia)
*''Ceropales carinitifrons'' Wahis, 1986 (Madagascar)
*''Ceropales angolaensis'' Móczar, 1989 (Angola)
*''Ceropales dayi'' Móczar, 1989 (Kenya)
*''Ceropales ferrugo'' Móczar, 1989 (Kenya)
*''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone)
*''Ceropales gessi'' Móczar, 1988 (South Africa)
*''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe)
*''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara)
*''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa)
*''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe)
*''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe)
*''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia)
*''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe)
*''Ceropales levipleuris'' Wahis, 1987 (Madagascar)
*''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal)
*''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region)
*''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia)
*''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda)
*''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa)
*''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe)
*''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa)
*''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania)
*''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo)
*''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal)
**''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal)
*''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa)
*''Ceropales seyrigi'' Wahis, 1987 (Madagascar)
*''Ceropales spinolai'' Móczar, 1988 (Guinea)
*''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel)
*''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania)
**''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo)
*''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda)
*''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe)
*''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia)
&lt;br&gt;
=== Genus ''Irenangelus'' ===
*''Irenangelus madescassus'' Wahis, 1988 (Madagascar)
&lt;br&gt;

==Eumeninae==
Photos of ''Antodynerus'' on GBIF:&lt;br&gt;
''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)&lt;br&gt; 
''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)&lt;br&gt;
''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)&lt;br&gt; 
''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)&lt;br&gt; 
''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)&lt;br&gt;
''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)&lt;br&gt;
''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)&lt;br&gt;

==Ants==
'''Subfamilies of Formicidae (WaspWeb)'''

Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23)
Amblyoponinae              7
Dolichoderinae           630
Dorylinae                 1 167
Formicinae            10 396       Camponotus   6 090; Lepisiota 1 046
Myrmicinae              8 484       Crematogaster  1 786;   Pheidole  1 468;   Messor  1 156 
Ponerinae                1 623
Proceratiinae                  3
Pseudomyrmecinae   296


Aenictinae One Afrotropical genus ''Aenictus'' &lt;br&gt;

Aenictogitoninae One Afrotropical genus ''Aenictogiton'' &lt;br&gt;

Amblyoponinae Five Afrotropical genera &lt;br&gt;

Apomyrminae One Afrotropical genus ''Apomyrma'' &lt;br&gt; 

Cerapachyinae Five Afrotropical genera&lt;br&gt;

Dolichoderinae Eight Afrotropical genera&lt;br&gt;

Dorylinae One Afrotropical genus ''Dorylus'' &lt;br&gt;

Formicinae 20 Afrotropical genera&lt;br&gt;

Leptanillinae One Afrotropical genus ''Leptanilla'' &lt;br&gt;

Myrmicinae 37 Afrotropical genera &lt;br&gt;

Ponerinae 18 Afrotropical genera &lt;br&gt;

Proceratiinae Three Afrotropical genera  &lt;br&gt;

Pseudomyrmecinae One Afrotropical genus Tetraponera &lt;br&gt;


&lt;gallery mode=packed heights=200&gt;
Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae 
Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae 
Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae 
Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae
Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae 
Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae  
Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae 
Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae 
AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae 
Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae 
Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae 
Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae 
Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae
&lt;/gallery&gt;

== N-P interactions ==
Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770.
'''Abstract'''
Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective.

==Diptera==
===Wing and leg-waving behavior in flies===
====Food detection====
*''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. 
**https://www.researchgate.net/publication/359760392
*It is also possible that some flies sample the air with the chemical sensors on their legs or feet.
**https://bugguide.net/node/view/217136/bgpage
====Courtship====
*Some Taeniapterinae are thought to wave their white-tipped front legs attract females.
**https://bugguide.net/node/view/217136/bgpage
*''Physiphora clausa'' appear to use leg-waving in courtship displays.
**https://www.flickr.com/photos/jean_hort/4663220062
*Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata''
**https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x
====Mimics for defense====
*Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae.
**https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ 
*Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging.
**https://www.jstor.org/stable/10.1086/674612
*Wing-waving to mimic salticid spiders.
**https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895&lt;br&gt;
&lt;br&gt;

===Number of iNat records in Acalyptrate fly families===
The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:&lt;br&gt;

* '''Carnoidea'''
** Acartophthalmidae         0
** Australimyzidae           0
** Braulidae (bee lice)         1
** Canacidae (beach flies)         3
** Carnidae (bird flies)         0
** Chloropidae (frit flies)         259
** Cryptochetidae              1
** Inbiomyiidae                0
** Milichiidae (freeloader flies) 158
&lt;br&gt;
* '''Diopsoidea'''
** Diopsidae (stalk-eyed flies)    545
** Gobryidae                   0
** Megamerinidae               0
** Nothybidae                   0
** Psilidae (rust flies)         29
** Somatiidae                   0
** Syringogastridae            0
&lt;br&gt;
* '''Ephydroidea'''
** Camillidae  0
** Campichoetidae   0
** Curtonotidae (quasimodo flies)   15
** Diastatidae 0
** Drosophilidae (vinegar and fruit flies) 312
** Ephydridae (shore flies) 117
&lt;br&gt;
* '''Lauxanioidea'''
** Celyphidae (beetle flies)  0
** Chamaemyiidae (aphid flies)  24
** Cremifaniidae  0
** Lauxaniidae (lauxaniid flies)  710
&lt;br&gt;
* '''Nerioidea'''
** Cypselosomatidae  0
** Fergusoninidae  0
** Micropezidae (stilt-legged flies)  245
** Neriidae  109
** Strongylophthalmyiidae  0
** Tanypezidae (stretched-foot flies)  0
&lt;br&gt;
* '''Opomyzoidea'''
** Agromyzidae (leaf-miner flies) 161
** Anthomyzidae  3
** Asteiidae  4
** Aulacigastridae  2
** Clusiidae (druid flies) 2
** Marginidae  0
** Neminidae 0 
** Neurochaetidae 0
** Odiniidae  0
** Opomyzidae  4
** Periscelididae  1
** Teratomyzidae  0
** Xenasteiidae  0
&lt;br&gt;
* '''Sciomyzoidea''' 
** Coelopidae (kelp flies)  51
** Conopidae (thick-headed flies)  192
** Dryomyzidae  1
** Helcomyzidae  0
** Helosciomyzidae  0
** Heterocheilidae  0
** Huttoninidae  0
** Natalimyzidae  0
** Phaeomyiidae  0
** Ropalomeridae  1
** Sciomyzidae (marsh flies)  67
** Sepsidae (black scavenger flies)  269
&lt;br&gt;
* '''Sphaeroceroidea'''
** Chyromyidae (golden flies)  19
** Heleomyzidae (heleomyzid flies)  151
** Nannodastiidae  0
** Sphaeroceridae (lesser dung flies)  48
&lt;br&gt;
* '''Tephritoidea'''
** Ctenostylidae  1
** Lonchaeidae (lance flies) 47
** Pallopteridae (flutter-wing flies)  5
** Piophilidae (cheese skipper flies) 1
** Platystomatidae (signal flies)  683
** Pyrgotidae (scarab-pursuing flies)  119
** Richardiidae  0
** Tachiniscidae  2
** Tephritidae (fruit flies)  1,759
** Ulidiidae (picture-winged flies) 165

== References ==</text>
      <sha1>ruh0yof32s5ebt687tpclpf6ewv1alr</sha1>
    </revision>
    <revision>
      <id>2819242</id>
      <parentid>2819241</parentid>
      <timestamp>2026-07-24T11:04:21Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Wings flavo-hyaline, often with a fusous (darker) apical margin */</comment>
      <origin>2819242</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19642" sha1="87s36w57zvd73yjlurz3zxqmwki0coo" xml:space="preserve">&lt;!--Info--&gt;
== Rough guide to Liris species of South Africa ==
=== Wings dark fuscous ===
Very dark wings, often with violet reflections
&lt;gallery mode=packed heights=200&gt;
Liris inaturalist 197329080.jpg

&lt;/gallery&gt;
=== Wings hyaline ===
Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin
&lt;gallery mode=packed heights=200&gt;


&lt;/gallery&gt;
=== Wings flavo-hyaline, often with a fusous (darker) apical margin ===
''Liris haemorrhoidalis'', ''Liris bembesianus'', ''Liris solstitialis''
&lt;gallery mode=packed heights=200&gt;
Liris on Crassula iN 42678436 03.jpg

&lt;/gallery&gt;

= Pompilidae of South Africa =
== South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips ==
&lt;gallery mode=packed heights=200&gt;
Inaturalist 258649905 b.jpg
Hemipepsis hilaris - inaturalist 10850475.jpg
Cyphononyx decipiens inat 26259647 b.jpg
Tachypompilus ignitus inaturalist 311015843 02.jpg
Pompilidae 2021 12 12 inaturalist 313386858 04.jpg
Pompilidae 2020 04 13 inaturalist 43563902 06.jpg
&lt;/gallery&gt;
*The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent.
*
&lt;br&gt;

== South African Pompilidae with fore-wings fuscous (black or very dark) ==
*The wings often have green-blue-violet reflections.
&lt;gallery mode=packed heights=200&gt;
Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus''
Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus''
Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus''
Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex''
Pompilidae_2019_05_28_0256.jpg|
Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp.
&lt;/gallery&gt;

&lt;br&gt;
=== Species  with black antennae, legs, head, thorax and abdomen ===
Some parts may be brown.

*''Java atropos''
*''Cyphononyx obscurus''
*''Hemipepsis vindex''
*''Hemipepsis vespertilio''
*''Hemipepsis braunsi''
*''Batozonellus fuliginosus''
&lt;br&gt;
=== Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red ===
*''Cyphononyx optimus''
*''Paracyphononyx zonatus''
&lt;br&gt;
&lt;br&gt;

== South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline ==
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg
Pompilidae inaturalist 46961473.jpg
Pompilidae iN 144781033 03.jpg

&lt;/gallery&gt;
*With fuscous (darker) wing apex
*One or two fuscous bands (faciated or bifaciated)
*Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted)


&lt;br&gt;
== South African Ceropalinae ==
Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020]
*''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia)
*''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region)
*''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa)
*''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe)
*''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa)
*''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?)
*''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) 
*''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe)
*''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda)
*''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa)
**''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe)
**''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa)
*''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa)
**= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319
*''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania)
*''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe)
&lt;br&gt;

==Afrotropical Ceropalinae ==

Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).&lt;ref name=Madl2020&gt;Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84.
[https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]&lt;/ref&gt;


Ceropalinae can be defined by:&lt;ref name=Brothers1993&gt;Brothers, D. J. &amp; Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. &amp; Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143&lt;/ref&gt;&lt;ref name=Waichert2015&gt; Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., &amp; Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]&lt;/ref&gt;

== Genera and species of Afrotropical Ceropalinae ==

This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa &amp; Schmid-Egger, 2025)&lt;ref name=CoL2025&gt; Kroupa, A. S., &amp; Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv &lt;/ref&gt; and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].&lt;br&gt;

=== Genus ''Ceropales'' ===
*''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia)
*''Ceropales angolaensis'' Móczar, 1989 (Angola)
*''Ceropales arnoldi'' Móczar, 1988 (Namibia)
*''Ceropales atra'' Móczar, 1991 (Botswana)
*''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region)
*''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia)
*''Ceropales carinitifrons'' Wahis, 1986 (Madagascar)
*''Ceropales angolaensis'' Móczar, 1989 (Angola)
*''Ceropales dayi'' Móczar, 1989 (Kenya)
*''Ceropales ferrugo'' Móczar, 1989 (Kenya)
*''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone)
*''Ceropales gessi'' Móczar, 1988 (South Africa)
*''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe)
*''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara)
*''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa)
*''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe)
*''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe)
*''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia)
*''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe)
*''Ceropales levipleuris'' Wahis, 1987 (Madagascar)
*''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal)
*''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region)
*''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia)
*''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda)
*''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa)
*''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe)
*''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa)
*''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania)
*''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo)
*''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal)
**''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal)
*''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa)
*''Ceropales seyrigi'' Wahis, 1987 (Madagascar)
*''Ceropales spinolai'' Móczar, 1988 (Guinea)
*''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel)
*''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania)
**''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo)
*''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda)
*''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe)
*''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia)
&lt;br&gt;
=== Genus ''Irenangelus'' ===
*''Irenangelus madescassus'' Wahis, 1988 (Madagascar)
&lt;br&gt;

==Eumeninae==
Photos of ''Antodynerus'' on GBIF:&lt;br&gt;
''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)&lt;br&gt; 
''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)&lt;br&gt;
''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)&lt;br&gt; 
''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)&lt;br&gt; 
''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)&lt;br&gt;
''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)&lt;br&gt;
''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)&lt;br&gt;

==Ants==
'''Subfamilies of Formicidae (WaspWeb)'''

Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23)
Amblyoponinae              7
Dolichoderinae           630
Dorylinae                 1 167
Formicinae            10 396       Camponotus   6 090; Lepisiota 1 046
Myrmicinae              8 484       Crematogaster  1 786;   Pheidole  1 468;   Messor  1 156 
Ponerinae                1 623
Proceratiinae                  3
Pseudomyrmecinae   296


Aenictinae One Afrotropical genus ''Aenictus'' &lt;br&gt;

Aenictogitoninae One Afrotropical genus ''Aenictogiton'' &lt;br&gt;

Amblyoponinae Five Afrotropical genera &lt;br&gt;

Apomyrminae One Afrotropical genus ''Apomyrma'' &lt;br&gt; 

Cerapachyinae Five Afrotropical genera&lt;br&gt;

Dolichoderinae Eight Afrotropical genera&lt;br&gt;

Dorylinae One Afrotropical genus ''Dorylus'' &lt;br&gt;

Formicinae 20 Afrotropical genera&lt;br&gt;

Leptanillinae One Afrotropical genus ''Leptanilla'' &lt;br&gt;

Myrmicinae 37 Afrotropical genera &lt;br&gt;

Ponerinae 18 Afrotropical genera &lt;br&gt;

Proceratiinae Three Afrotropical genera  &lt;br&gt;

Pseudomyrmecinae One Afrotropical genus Tetraponera &lt;br&gt;


&lt;gallery mode=packed heights=200&gt;
Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae 
Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae 
Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae 
Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae
Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae 
Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae  
Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae 
Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae 
AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae 
Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae 
Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae 
Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae 
Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae
&lt;/gallery&gt;

== N-P interactions ==
Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770.
'''Abstract'''
Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective.

==Diptera==
===Wing and leg-waving behavior in flies===
====Food detection====
*''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. 
**https://www.researchgate.net/publication/359760392
*It is also possible that some flies sample the air with the chemical sensors on their legs or feet.
**https://bugguide.net/node/view/217136/bgpage
====Courtship====
*Some Taeniapterinae are thought to wave their white-tipped front legs attract females.
**https://bugguide.net/node/view/217136/bgpage
*''Physiphora clausa'' appear to use leg-waving in courtship displays.
**https://www.flickr.com/photos/jean_hort/4663220062
*Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata''
**https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x
====Mimics for defense====
*Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae.
**https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ 
*Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging.
**https://www.jstor.org/stable/10.1086/674612
*Wing-waving to mimic salticid spiders.
**https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895&lt;br&gt;
&lt;br&gt;

===Number of iNat records in Acalyptrate fly families===
The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:&lt;br&gt;

* '''Carnoidea'''
** Acartophthalmidae         0
** Australimyzidae           0
** Braulidae (bee lice)         1
** Canacidae (beach flies)         3
** Carnidae (bird flies)         0
** Chloropidae (frit flies)         259
** Cryptochetidae              1
** Inbiomyiidae                0
** Milichiidae (freeloader flies) 158
&lt;br&gt;
* '''Diopsoidea'''
** Diopsidae (stalk-eyed flies)    545
** Gobryidae                   0
** Megamerinidae               0
** Nothybidae                   0
** Psilidae (rust flies)         29
** Somatiidae                   0
** Syringogastridae            0
&lt;br&gt;
* '''Ephydroidea'''
** Camillidae  0
** Campichoetidae   0
** Curtonotidae (quasimodo flies)   15
** Diastatidae 0
** Drosophilidae (vinegar and fruit flies) 312
** Ephydridae (shore flies) 117
&lt;br&gt;
* '''Lauxanioidea'''
** Celyphidae (beetle flies)  0
** Chamaemyiidae (aphid flies)  24
** Cremifaniidae  0
** Lauxaniidae (lauxaniid flies)  710
&lt;br&gt;
* '''Nerioidea'''
** Cypselosomatidae  0
** Fergusoninidae  0
** Micropezidae (stilt-legged flies)  245
** Neriidae  109
** Strongylophthalmyiidae  0
** Tanypezidae (stretched-foot flies)  0
&lt;br&gt;
* '''Opomyzoidea'''
** Agromyzidae (leaf-miner flies) 161
** Anthomyzidae  3
** Asteiidae  4
** Aulacigastridae  2
** Clusiidae (druid flies) 2
** Marginidae  0
** Neminidae 0 
** Neurochaetidae 0
** Odiniidae  0
** Opomyzidae  4
** Periscelididae  1
** Teratomyzidae  0
** Xenasteiidae  0
&lt;br&gt;
* '''Sciomyzoidea''' 
** Coelopidae (kelp flies)  51
** Conopidae (thick-headed flies)  192
** Dryomyzidae  1
** Helcomyzidae  0
** Helosciomyzidae  0
** Heterocheilidae  0
** Huttoninidae  0
** Natalimyzidae  0
** Phaeomyiidae  0
** Ropalomeridae  1
** Sciomyzidae (marsh flies)  67
** Sepsidae (black scavenger flies)  269
&lt;br&gt;
* '''Sphaeroceroidea'''
** Chyromyidae (golden flies)  19
** Heleomyzidae (heleomyzid flies)  151
** Nannodastiidae  0
** Sphaeroceridae (lesser dung flies)  48
&lt;br&gt;
* '''Tephritoidea'''
** Ctenostylidae  1
** Lonchaeidae (lance flies) 47
** Pallopteridae (flutter-wing flies)  5
** Piophilidae (cheese skipper flies) 1
** Platystomatidae (signal flies)  683
** Pyrgotidae (scarab-pursuing flies)  119
** Richardiidae  0
** Tachiniscidae  2
** Tephritidae (fruit flies)  1,759
** Ulidiidae (picture-winged flies) 165

== References ==</text>
      <sha1>87s36w57zvd73yjlurz3zxqmwki0coo</sha1>
    </revision>
  </page>
  <page>
    <title>Maritime Health Research and Education-NET</title>
    <ns>0</ns>
    <id>267737</id>
    <revision>
      <id>2819148</id>
      <parentid>2817902</parentid>
      <timestamp>2026-07-23T15:30:03Z</timestamp>
      <contributor>
        <username>NDM2024</username>
        <id>2984088</id>
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      <text bytes="5475" sha1="q8qsbe66d6xg94kwu0ugd7n4rrpbjaa" xml:space="preserve">== '''The John Snow Prediabetes Institute.''' ==
[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]]

[[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]
Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]]
[[File:John Snow.jpg||50px|left| John Snow in the early nineteenth century]]
[[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]]

1. '&lt;nowiki/&gt;'''Prevalence studies''''

1.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

1.2 The Maritime Officer`s Health-Database &lt;ref&gt;https://www.dropbox.com/t/8LjP7cmulhr2x8Ty &lt;/ref&gt;

1.3 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

1.4. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

1.5. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;


2. '''Intervention studies''' Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

- General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

- John Snow Institute bylaws &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

3. '''Publications and pptx''' 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;

4. '''Strategies for research and implementation''' 
For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools..
A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:

'''5. Minutes from meetings''' &lt;ref&gt; https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 &lt;/ref&gt;

6. '''Prediabetes-Remission Research Network:'''
&lt;small&gt;Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,&lt;/small&gt;
==References==
[[Category:Prediabetes ]]
&lt;references /&gt;Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
      <sha1>q8qsbe66d6xg94kwu0ugd7n4rrpbjaa</sha1>
    </revision>
    <revision>
      <id>2819149</id>
      <parentid>2819148</parentid>
      <timestamp>2026-07-23T15:31:22Z</timestamp>
      <contributor>
        <username>NDM2024</username>
        <id>2984088</id>
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      <text bytes="5388" sha1="gccwhln9jm6rt71hlb6qx37ngrl9soq" xml:space="preserve">== '''The John Snow Prediabetes Institute.''' ==
[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]]

[[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]
Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]]
[[File:John Snow.jpg||150px|left| John Snow in the early nineteenth century]]
[[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]]

1. '&lt;nowiki/&gt;'''Prevalence studies''''

1.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

1.2 The Maritime Officer`s Health-Database &lt;ref&gt;https://www.dropbox.com/t/8LjP7cmulhr2x8Ty &lt;/ref&gt;

1.3 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

1.4. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

1.5. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;


2. '''Intervention studies''' Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

- General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

- Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

- John Snow Institute bylaws &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

3. '''Publications and pptx''' 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;&lt;ref name=&quot;:0&quot;&gt; https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf &lt;/ref&gt;

4. '''Strategies for research and implementation''' 
For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools..
A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:

5. '''Prediabetes-Remission Research Network:'''
&lt;small&gt;Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,&lt;/small&gt;
==References==
[[Category:Prediabetes ]]
&lt;references /&gt;Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
&lt;references /&gt;</text>
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  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2819140</id>
      <parentid>2819046</parentid>
      <timestamp>2026-07-23T13:44:25Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2819140</origin>
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      <text bytes="9878" sha1="3akmfzws7ksknn3gynzog33nhy4pyvj" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260723.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
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&lt;noinclude&gt;The [[Motivation and emotion/Book|motivation and emotion book]]&lt;/noinclude&gt;&lt;includeonly&gt;This book&lt;/includeonly&gt; consists of over 1,800 online book chapters about how psychological science can be used to improve human lives. This work is published via [[Wikiversity]], an open wiki platform supported by the [[Wikimedia Foundation]]. Third year undergraduate psychology students develop the chapters as a learning and assessment exercise for the [[Motivation and emotion|motivation and emotion]] unit at the [[University of Canberra]], Australia, to engage students in collaborative authorship. Using [[open academia|open academic]] principles, [[emerging scholars]] create [[Open Educational Resources|open educational resources]] to develop and demonstrate their capacity to apply psychological theory and research to real-world problems.&lt;noinclude&gt;

{{RoundBoxTop}}
&lt;div align=&quot;center&quot;&gt;Book theme:&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''Understanding and improving our motivational and emotional lives using psychological science''&lt;/div&gt;
{{RoundBoxBottom}}

Learn more about this project from an educational psychology perspective:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki: An open education case study]] (Neill, 2024)
* [[User:Jtneill/Publications/Wikis provide a rich environment for collaborative open educational practices: Motivation and emotion case study|Wikis provide a rich environment for collaborative open educational practices]] (Neill, 2024)
&lt;!--
Also of interest may be the:&lt;br&gt;
[https://x.com/jtneill/status/1422496816709312513 2021 book chapter highlights Twitter thread].
--&gt;
[[Category:Motivation and emotion/Book]]&lt;/noinclude&gt;</text>
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&lt;noinclude&gt;The [[Motivation and emotion/Book|motivation and emotion book]]&lt;/noinclude&gt;&lt;includeonly&gt;This book&lt;/includeonly&gt; consists of over 1,800 online book chapters about how psychological science can be used to improve human lives. &lt;noinclude&gt;This&lt;/noinclude&gt;&lt;includeonly&gt;The&lt;/includeonly&gt; work is published via [[Wikiversity]], an open wiki platform supported by the [[Wikimedia Foundation]]. Third year undergraduate psychology students develop the chapters as a learning and assessment exercise for the [[Motivation and emotion|motivation and emotion]] unit at the [[University of Canberra]], Australia, to engage students in collaborative authorship. Using [[open academia|open academic]] principles, [[emerging scholars]] create [[Open Educational Resources|open educational resources]] to develop and demonstrate their capacity to apply psychological theory and research to real-world problems.&lt;noinclude&gt;

{{RoundBoxTop}}
&lt;div align=&quot;center&quot;&gt;Book theme:&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''Understanding and improving our motivational and emotional lives using psychological science''&lt;/div&gt;
{{RoundBoxBottom}}

Learn more about this project from an educational psychology perspective:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki: An open education case study]] (Neill, 2024)
* [[User:Jtneill/Publications/Wikis provide a rich environment for collaborative open educational practices: Motivation and emotion case study|Wikis provide a rich environment for collaborative open educational practices]] (Neill, 2024)
&lt;!--
Also of interest may be the:&lt;br&gt;
[https://x.com/jtneill/status/1422496816709312513 2021 book chapter highlights Twitter thread].
--&gt;
[[Category:Motivation and emotion/Book]]&lt;/noinclude&gt;</text>
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&lt;noinclude&gt;The [[Motivation and emotion/Book|motivation and emotion book]]&lt;/noinclude&gt;&lt;includeonly&gt;This book&lt;/includeonly&gt; consists of over 1,800 online book chapters about how psychological science can be used to improve human lives. &lt;noinclude&gt;This&lt;/noinclude&gt;&lt;includeonly&gt;The&lt;/includeonly&gt; work is published via [[Wikiversity]], an open wiki platform supported by the [[Wikimedia Foundation]]. Third year undergraduate psychology students develop the chapters as a learning and assessment exercise for the [[Motivation and emotion|motivation and emotion]] unit at the [[University of Canberra]], Australia. Using [[open academia|open academic]] principles to engage students in collaborative authorship, [[emerging scholars]] create [[Open Educational Resources|open educational resources]] to develop and demonstrate their capacity to apply psychological theory and research to real-world problems.&lt;noinclude&gt;

{{RoundBoxTop}}
&lt;div align=&quot;center&quot;&gt;Book theme:&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''Understanding and improving our motivational and emotional lives using psychological science''&lt;/div&gt;
{{RoundBoxBottom}}

Learn more about this project from an educational psychology perspective:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki: An open education case study]] (Neill, 2024)
* [[User:Jtneill/Publications/Wikis provide a rich environment for collaborative open educational practices: Motivation and emotion case study|Wikis provide a rich environment for collaborative open educational practices]] (Neill, 2024)
&lt;!--
Also of interest may be the:&lt;br&gt;
[https://x.com/jtneill/status/1422496816709312513 2021 book chapter highlights Twitter thread].
--&gt;
[[Category:Motivation and emotion/Book]]&lt;/noinclude&gt;</text>
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&lt;noinclude&gt;The [[Motivation and emotion/Book|motivation and emotion book]]&lt;/noinclude&gt;&lt;includeonly&gt;This book&lt;/includeonly&gt; consists of over 1,800 online book chapters about how psychological science can be used to improve human lives. &lt;noinclude&gt;This&lt;/noinclude&gt;&lt;includeonly&gt;The&lt;/includeonly&gt; work is published via [[Wikiversity]], an open wiki platform supported by the [[Wikimedia Foundation]]. Third year undergraduate psychology students develop the chapters as a learning and assessment exercise for the [[Motivation and emotion|motivation and emotion]] unit at the [[University of Canberra]], Australia. Using [[open academia|open academic]] principles to engage students in collaborative authorship, [[emerging scholars]] create these [[Open Educational Resources|open educational resources]] to develop and demonstrate their capacity to apply psychological theory and research to real-world problems.&lt;noinclude&gt;

{{RoundBoxTop}}
&lt;div align=&quot;center&quot;&gt;Book theme:&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''Understanding and improving our motivational and emotional lives using psychological science''&lt;/div&gt;
{{RoundBoxBottom}}

Learn more about this project from an educational psychology perspective:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki: An open education case study]] (Neill, 2024)
* [[User:Jtneill/Publications/Wikis provide a rich environment for collaborative open educational practices: Motivation and emotion case study|Wikis provide a rich environment for collaborative open educational practices]] (Neill, 2024)
&lt;!--
Also of interest may be the:&lt;br&gt;
[https://x.com/jtneill/status/1422496816709312513 2021 book chapter highlights Twitter thread].
--&gt;
[[Category:Motivation and emotion/Book]]&lt;/noinclude&gt;</text>
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]]. Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] which include a public link to the conversation or the tool used and prompt details. [[w:Fact-checking|Fact-check]] genAI content and only cite peer-reviewed sources which you have consulted. Human-rewrite genAI content to enhance quality.&lt;includeonly&gt; [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid but should not replace independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for low-level tasks such as improving spelling and grammar or using Studiosity Writing Feedback+.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
;Use ethically, with caution
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

;How to acknowledge
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing too much acknowledgement rather than not enough. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

;Limitations
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

;Fact-check and cite
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

;Going forth
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work. Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]]. Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] which include a public link to the conversation or the tool used and prompt details. [[w:Fact-checking|Fact-check]] genAI content and only cite peer-reviewed sources which you have consulted. Human-rewrite genAI content to enhance quality.&lt;includeonly&gt; [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid but should not replace independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for low-level tasks such as improving spelling and grammar or using Studiosity Writing Feedback+.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
;Use ethically, with caution
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

;How to acknowledge
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing too much acknowledgement rather than not enough. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

;Limitations
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

;Fact-check and cite
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

;Going forth
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work. Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
&lt;/noinclude&gt;</text>
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]]. Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] which include a public link to the conversation or the tool used and prompt details. [[w:Fact-checking|Fact-check]] genAI content and only cite peer-reviewed sources which you have consulted. Human-rewrite genAI content to enhance quality.&lt;includeonly&gt; [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for low-level tasks such as improving spelling and grammar or using Studiosity Writing Feedback+.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
;Use ethically, with caution
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

;How to acknowledge
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing too much acknowledgement rather than not enough. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

;Limitations
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

;Fact-check and cite
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

;Going forth
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work. Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
&lt;/noinclude&gt;</text>
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]]. Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] which include a public link to the conversation or the tool used and prompt details. [[w:Fact-checking|Fact-check]] genAI content and only cite peer-reviewed sources which you have consulted. Human-rewrite genAI content to enhance quality.&lt;includeonly&gt; [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for low-level tasks such as improving spelling and grammar or using Studiosity Writing Feedback+.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
;Use ethically, with caution
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

;How to use
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

Refine prompts based on critical reading for better results. 

;How to acknowledge
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing too much acknowledgement rather than not enough. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

;Limitations
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

;Fact-check and cite
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

;Going forth
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
&lt;/noinclude&gt;</text>
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]]. Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] which include a public link to the conversation or the tool used and prompt details. [[w:Fact-checking|Fact-check]] genAI content and only cite peer-reviewed sources which you have consulted. Human-rewrite genAI content to enhance quality.&lt;includeonly&gt; [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for low-level tasks such as improving spelling and grammar or using Studiosity Writing Feedback+.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
&lt;/noinclude&gt;</text>
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__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
* '''Permitted''' - The ethical use of GenAI is allowed in completing the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]].
* '''Restricted''' - The use of GenAI is NOT allowed in completing the [[Motivation and emotion/Assessment/Exam|exam]].

To use genAI appropriately:
* Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|
Wikiversity edit summaries]] with a publicly accessible link to the conversation or tool used and prompt details.
* [[w:Fact-checking|Fact-check]] genAI content.
* Only cite peer-reviewed sources which you have consulted.
* Human-rewrite genAI content to enhance quality.&lt;includeonly&gt;
* [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

[[w:Generative AI|Generative artificial intelligence]] (genAI) use is '''permitted''' for the [[Motivation and emotion/Assessment/Major project|major project]] (topic development and book chapter) and '''restricted''' for the [[Motivation and emotion/Assessment/Exam|exam]].

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid, but should not replace, independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for low-level tasks such as improving spelling and grammar or using Studiosity Writing Feedback+.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources, with human rewriting to enhance quality. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
===Use ethically, with caution===
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously, as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

===How to use===
Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* correcting spelling and grammar
* image generation (e.g., see Figure 1)
* scenario development
* critical feedback and suggestions for improvement

===How to acknowledge===
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be used as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing acknowledgement. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling or using Studiosity Writing Feedback+.

===Limitations===
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort reading and prompting will yield low quality results. Refine prompts based on critical reading and thinking to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

===Fact-check and cite===
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Guide and craft genAI responses based on your reading of peer-reviewed theory and research. Low-energy or unreflective reuse of genAI text without further investigation and human-writing based on reviewing primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. 

===Going forth===
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work.

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
&lt;/noinclude&gt;</text>
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  <page>
    <title>24-cell</title>
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      <text bytes="240528" sha1="lt9pfocf027hsolbcktpwe67upc1imj" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240529" sha1="tj4edjpc6ijqz7e2o2cvems4n8morck" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
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==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
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|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240527" sha1="5416qgzxtw1awfq2z0swg5vf9sa6zpq" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240534" sha1="8llj8ohvwu12h197ptgd0dho901n16g" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240534" sha1="7qwhth6one7cc3akon422trxij0196s" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240524" sha1="7lute7n21dq2upyxwmatxgwu8dh8tfu" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240524" sha1="1n1rfo0mpohl2ndfpiyhrjrgw1oauhu" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
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|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|
|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240524" sha1="ixkqqz9w17kaq5oohk9vbg7vj0sfqnb" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240518" sha1="da8bjyyo5qt5zm5z76yrnot01lho0gn" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240519" sha1="in28d9q6gbqlgns64bvbyxowetu1gzy" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240512" sha1="8gx5q4wdm8q0f1o26kzrj9wa5yt3ob8" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240512" sha1="t9nmgjmsdw47oi0tv4r61ab9az0n11z" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240514" sha1="5sr0l2wn73t0bwyl7rxejvx5xn7kszm" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240512" sha1="t9nmgjmsdw47oi0tv4r61ab9az0n11z" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240514" sha1="s3noaipv41wa58c5cs6aosxviiazxjz" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240512" sha1="t9nmgjmsdw47oi0tv4r61ab9az0n11z" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240515" sha1="5660n75gt4hwqgb4usy3z4isqit3fjk" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240514" sha1="f0fws4ep7urz6rlducbc7tjb4isu9ti" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240520" sha1="t2gdknmfkprosl0dkv2fowytc928vt0" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240515" sha1="49te9nerdj2lbtourld109xcqjo40w8" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240515" sha1="495iqdlsitps7n75dxbotl4vp56kmh9" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240515" sha1="49te9nerdj2lbtourld109xcqjo40w8" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
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|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|
|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240512" sha1="5dsewpejfj34uur7208eal655izfy0t" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240515" sha1="pmhjmejlyzma4b511md768fwi8gdnd6" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240513" sha1="3haytxnsauyybibsa1czywozpnbuqg7" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240515" sha1="pmhjmejlyzma4b511md768fwi8gdnd6" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240504" sha1="7s3g8zvfxe15i9u35ys38zjebna1jut" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
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|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|
|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240501" sha1="1gdf0y9wmirx6stx9tqsclfohaenxcp" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240509" sha1="r1vdw8kozegzj572hyfooc03ga2ooql" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240517" sha1="6osvb1uw5iqrcm49g68hb020tubx940" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240501" sha1="kuj7j14ogeem113rwroll41baszy0ds" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
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==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
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|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240498" sha1="5i27r6u7aji87ffiufdx11hb5gf2aq3" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240444" sha1="b0gwg8zdpr7r16vj1d0n7hpygtl90ys" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons||100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240498" sha1="5i27r6u7aji87ffiufdx11hb5gf2aq3" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="242384" sha1="9d3mepo51w62z98c8ighdguew2jodo3" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240498" sha1="32qmqxiha3hdid0ll6e11k7axdzys4q" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240492" sha1="c79waoxwncnlynt3tji4ay6e0ktvdwt" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240521" sha1="qv2vuaft61l63y69ey9imv2icbktsmp" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column.

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[8] 10𝝅 {12/5}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]&lt;br&gt;[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}?
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]&lt;br&gt;[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[12] 1𝝅 {2}?
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]&lt;br&gt;[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[24] 0𝝅 {1}?
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]&lt;br&gt;[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], &lt;s&gt;passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each&lt;/s&gt;.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, &lt;s&gt;each having one edge in each great hexagon plane, and&lt;/s&gt; circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines.

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
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      <text bytes="1388" sha1="dwpgwq18rhs7y8m3ij81s69g7ivxr3d" xml:space="preserve">&lt;noinclude&gt;==Alternative assessment==&lt;/noinclude&gt;
* The [[Motivation and emotion/Assessment/Major project|major project]] ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]) involves developing and publishing work on [[Main page|Wikiversity]], a collaborative, online, public platform
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[[Category:Motivation and emotion/Assessment]]
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      <text bytes="1375" sha1="2a57ralhodz7jmycixhjlt1bvtdkaye" xml:space="preserve">&lt;noinclude&gt;==Alternative assessment==&lt;/noinclude&gt;
* The [[Motivation and emotion/Assessment/Major project|major project]] ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]) involves developing and publishing work on [[Main page|Wikiversity]], a collaborative, online, public platform.
* Students who have online privacy concerns are encouraged to use an anonymous account name.
* [https://policies.canberra.edu.au/document/view-current.php?id=108&amp;version=1#major2 Students own the intellectual property of their work]. Contributing to Wikiversity requires [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0] licensing, which is irrevocable.
* Alternative assessment that meets the [[Motivation and emotion/About/Learning outcomes|learning outcomes]] and [[Motivation and emotion/About/Graduate attributes|graduate attributes]] may be negotiated with the [[Motivation and emotion/About/Staff|unit convener]] where reasonable grounds are presented.
* Students who wish to request alternative assessment should contact the [[Motivation and emotion/About/Staff|unit convener]] by email as early as possible. If no request is made, participation in the standard assessments tasks is assumed&lt;noinclude&gt;
[[Category:Motivation and emotion/Assessment]].
&lt;/noinclude&gt;</text>
      <sha1>2a57ralhodz7jmycixhjlt1bvtdkaye</sha1>
    </revision>
    <revision>
      <id>2819233</id>
      <parentid>2819213</parentid>
      <timestamp>2026-07-24T07:37:35Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Update for 2026</comment>
      <origin>2819233</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1353" sha1="7gpql2fo7ltn7f7w7u25qim3z4m58qh" xml:space="preserve">&lt;noinclude&gt;==Alternative assessment==&lt;/noinclude&gt;
* The [[Motivation and emotion/Assessment/Major project|major project]] ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]) uses [[Main page|Wikiversity]], a collaborative, online, public platform.
* Use an anonymous Wikiversity account if you have privacy concern.
* [https://policies.canberra.edu.au/document/view-current.php?id=108&amp;version=1#major2 Students retain the intellectual property of their work]. Contributing to Wikiversity requires the work to be licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 license], which is irrevocable.
* Students who have reasonable grounds for not completing the Topic Development and Book Chapter assessments on Wikiversity may request an alternative assessment that addresses the same [[Motivation and emotion/About/Learning outcomes|learning outcomes]] and [[Motivation and emotion/About/Graduate attributes|graduate attributes]].
* Requests should be made by email to the [[Motivation and emotion/About/Staff|Unit Convener]] as early as possible. If no request is made, students are expected to complete the standard Wikiversity-based assessments.&lt;noinclude&gt;
[[Category:Motivation and emotion/Assessment]].
&lt;/noinclude&gt;</text>
      <sha1>7gpql2fo7ltn7f7w7u25qim3z4m58qh</sha1>
    </revision>
  </page>
  <page>
    <title>Module:Params</title>
    <ns>828</ns>
    <id>308015</id>
    <revision>
      <id>2819136</id>
      <parentid>2818395</parentid>
      <timestamp>2026-07-23T13:31:20Z</timestamp>
      <contributor>
        <username>Grufo</username>
        <id>1192007</id>
      </contributor>
      <comment>Update from [[d:Special:GoToLinkedPage/mediawikiwiki/Q122696746|master]] using [[mw:Synchronizer| #Synchronizer]]</comment>
      <origin>2819136</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="88036" sha1="s0wwht3ty2l3u443tnfm9mh3z5pmpgj" xml:space="preserve">require[[strict]]


	---                                        ---
	---     PRIVATE ENVIRONMENT                ---
	---    ________________________________    ---
	---                                        ---


	--[[ Abstract utilities ]]--
	----------------------------


-- Helper function for `string.gsub()` (for managing zero-padded numbers)
local function zero_padded (str)
	return ('%03d%s'):format(#str, str)
end


-- Helper function for `table.sort()` (for natural sorting)
local function natural_sort (var1, var2)
	return var1:gsub('%d+', zero_padded) &lt; var2:gsub('%d+', zero_padded)
end


-- Return a copy or a reference to a table
local function copy_or_ref_table (src, refonly)
	if refonly then return src end
	local newtab = {}
	for key, val in pairs(src) do newtab[key] = val end
	return newtab
end


-- Copy at most N items (of all kinds) from `src` to `dest` and return `dest`
local function copy_table_maxn (dest, src, len)
	local idx = 1
	for key, val in pairs(src) do
		dest[key], idx = val, idx + 1
		if idx &gt; len then break end
	end
	return dest
end


-- Remove some numeric elements from a table, shifting everything to the left
local function remove_numeric_keys (tbl, idx, len)
	local cache, tmp = {}, idx + len - 1
	for key, val in pairs(tbl) do
		if type(key) == 'number' and key &gt;= idx then
			if key &gt; tmp then cache[key - len] = val end
			tbl[key] = nil
		end
	end
	for key, val in pairs(cache) do tbl[key] = val end
end


-- Make a reduced copy of a table (shifting in both directions if necessary)
local function copy_table_reduced (tbl, idx, len)
	local ret, tmp = {}, idx + len - 1
	if idx &gt; 0 then
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &lt; idx then
				ret[key] = val
			elseif key &gt; tmp then ret[key - len] = val end
		end
	elseif tmp &gt; 0 then
		local nshift = 1 - idx
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' then ret[key] = val
			elseif key &gt; tmp then ret[key - tmp] = val
			elseif key &lt; idx then ret[key + nshift] = val end
		end
	else
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &gt; tmp then
				ret[key] = val
			elseif key &lt; idx then ret[key + len] = val end
		end
	end
	return ret
end


-- Make an expanded copy of a table (shifting in both directions if necessary)
local function copy_table_expanded (tbl, idx, len)
	local ret, tmp = {}, idx + len - 1
	if idx &gt; 0 then
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &lt; idx then
				ret[key] = val
			else ret[key + len] = val end
		end
	elseif tmp &gt; 0 then
		local nshift = idx - 1
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' then ret[key] = val
			elseif key &gt; 0 then ret[key + tmp] = val
			elseif key &lt; 1 then ret[key + nshift] = val end
		end
	else
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &gt; tmp then
				ret[key] = val
			else ret[key - len] = val end
		end
	end
	return ret
end


-- Given a table, create two new tables containing the sorted list of keys
local function get_key_list_sorted (tbl, sort_fn)
	local nums, words, nn, nw = {}, {}, 0, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			nn = nn + 1
			nums[nn] = key
		else
			nw = nw + 1
			words[nw] = key
		end
	end
	table.sort(nums)
	table.sort(words, sort_fn)
	return nums, words, nn, nw
end


-- Parse a parameter name string and return it as a string or a number
local function get_parameter_name (par_str)
	local ret = par_str:match'^%s*(.-)%s*$'
	if ret ~= '0' and ret:find'^%-?[1-9]%d*$' == nil then return ret end
	return tonumber(ret)
end


-- Move a key from a table to another, but only if under a different name and
-- always parsing numeric strings as numbers
local function steal_if_renamed (val, src, skey, dest, dkey)
	local realkey = get_parameter_name(dkey)
	if skey ~= realkey then dest[realkey], src[skey] = val, nil end
end


	--[[ Public strings ]]--
	------------------------


-- Special match keywords (functions and modifiers MUST avoid these names)
local mkeywords = {
	['or'] = 0,
	pattern = 1,
	plain = 2,
	strict = 3
}


-- Sort functions (functions and modifiers MUST avoid these names)
local sortfunctions = {
	alphabetically = false,
	naturally = natural_sort
}


-- Callback styles for the `mapping_*` and `renaming_*` class of modifiers
-- (functions and modifiers MUST avoid these names)
--[[

Meanings of the columns:

  col[1] = Loop type (0-3)
  col[2] = Number of module arguments that the style requires (1-3)
  col[3] = Minimum number of sequential parameters passed to the callback
  col[4] = Name of the callback parameter where to place each parameter name
  col[5] = Name of the callback parameter where to place each parameter value
  col[6] = Argument in the modifier's invocation that will override `col[4]`
  col[7] = Argument in the modifier's invocation that will override `col[5]`

A value of `-1` indicates that no meaningful value is stored (i.e. `nil`)

]]--
local mapping_styles = {
	names_and_values = { 3, 2, 2, 1, 2, -1, -1 },
	values_and_names = { 3, 2, 2, 2, 1, -1, -1 },
	values_only = { 1, 2, 1, -1, 1, -1, -1 },
	names_only = { 2, 2, 1, 1, -1, -1, -1 },
	names_and_values_as = { 3, 4, 0, -1, -1, 2, 3 },
	names_only_as = { 2, 3, 0, -1, -1, 2, -1 },
	values_only_as = { 1, 3, 0, -1, -1, -1, 2 },
	blindly = { 0, 2, 0, -1, -1, -1, -1 }
}


-- Memory slots (functions and modifiers MUST avoid these names)
local memoryslots = {
	h = 'header',
	f = 'footer',
	i = 'itersep',
	l = 'lastsep',
	n = 'ifngiven',
	p = 'pairsep',
	s = 'oxfordsep'
}


-- Possible trimming modes for the `parsing` modifier
local trim_parse_opts = {
	trim_none = { false, false },
	trim_positional = { false, true },
	trim_named = { true, false },
	trim_all = { true, true }
}


-- Possible string modes for the iteration separator in the `parsing` and
-- `reinterpreting` modifiers
local isep_parse_opts = {
	splitter_pattern = false,
	splitter_string = true
}


-- Possible string modes for the key-value separator in the `parsing` and
-- `reinterpreting` modifiers
local psep_parse_opts = {
	setter_pattern = false,
	setter_string = true
}


-- Possible position references for the `splicing` modifier
local position_references = {
	add_nothing = 0,
	add_smallest_number = 1,
	add_last_of_sequence = 2,
	add_largest_number = 3
}


-- Possible modes for the `reassigning` modifier
local a_modes = {
	transfer = 0,
	clone = 1,
	rename = 2,
	copy = 3,
	sacrifice = 4,
	provide = 5,
	spare = 7
}


-- Functions and modifiers MUST avoid these names too: `here`, `in_substack`,
-- `let`, `expose`, `use`, `with_flushed_glue`, `without_flushed_glue`
-- `without_sorting`


	--[[ Private constants ]]--
	---------------------------


-- Hard-coded name of the module (to avoid going through `frame:getTitle()`)
local modulename = 'Module:Params'


-- The functions listed here declare that they don't need the `frame.args`
-- metatable to be copied into a regular table; if they are modifiers they also
-- guarantee that they will make their own (modified) copy available
local refpipe = {
	call_for_each_group = true,
	--coins = true,
	count = true,
	evaluating = true,
	for_each = true,
	list = true,
	list_values = true,
	list_maybe_with_names = true,
	value_of = true
}


-- The functions listed here declare that they don't need the
-- `frame:getParent().args` metatable to be copied into a regular table; if 
-- they are modifiers they also guarantee that they will make their own
-- (modified) copy available
local refparams = {
	call_for_each_group = true,
	combining = true,
	combining_by_calling = true,
	combining_values = true,
	concat_and_call = true,
	concat_and_invoke = true,
	concat_and_magic = true,
	count = true,
	grouping_by_calling = true,
	mixing_names_and_values = true,
	keeping_at_most = true,
	renaming_by_mixing = true,
	renaming_to_sequence = true,
	renaming_to_uppercase = true,
	renaming_to_lowercase = true,
	--renaming_to_values = true,
	shifting = true,
	splicing = true,
	--swapping_names_and_values = true,
	value_of = true,
	with_name_matching = true
}


-- Maximum number of numeric parameters that can be filled, if missing (we
-- chose an arbitrary number for this constant; you can discuss about its
-- optimal value at Module talk:Params)
local maxfill = 1024


-- The private table of functions
local library = {}


-- Functions and modifiers that can only be invoked in first position
local static_iface = {}


	--[[ Private functions ]]--
	---------------------------


-- Create a new context
local function context_new (child_frame)
	local main_frame = child_frame:getParent()
	return {
		frame = main_frame,
		opipe = child_frame.args,
		oparams = main_frame.args,
		firstposonly = static_iface,
		iterfunc = pairs,
		sorttype = 0,
		n_parents = 0,
		n_children = 0,
		n_available = maxfill
	}
end


-- Move to the next action within the user-given list
local function context_iterate (ctx, n_forward)
	local nextfn
	if ctx.pipe[n_forward] ~= nil then
		nextfn = ctx.pipe[n_forward]:match'^%s*(.*%S)'
	end
	if nextfn == nil then error(modulename ..
		': You must specify a function to call', 0) end
	if library[nextfn] == nil then
		if ctx.firstposonly[nextfn] == nil then error(modulename ..
			': The function ‘' .. nextfn .. '’ does not exist', 0)
		else error(modulename .. ': The ‘' .. nextfn ..
			'’ directive can only appear in first position', 0)
		end
	end
	remove_numeric_keys(ctx.pipe, 1, n_forward)
	return library[nextfn]
end


-- Main loop
local function main_loop (ctx, start_with)
	local fn = start_with
	repeat fn = fn(ctx) until not fn
	if ctx.n_parents &gt; 0 then error(modulename ..
		': One or more ‘merging_substack’ directives are missing', 0) end
	if ctx.n_children &gt; 0 then error(modulename ..
		', For some of the snapshots either the ‘flushing’ directive is missing or a group has not been properly closed with ‘merging_substack’', 0) end
end


-- Load a `setting`-like directive string into the `dest` table
local function set_strings_from_opts (dest, opts, start_from)
	local cmd
	if opts[start_from] == nil then return start_from - 1 end
	cmd = opts[start_from]:gsub('%s+', ''):gsub('/+', '/')
		:match'^/*(.*[^/])'
	if cmd == nil then return start_from end
	local vname, chr
	local amap, sep, argc = {}, string.byte('/'), start_from + 1
	for idx = 1, #cmd do
		chr = cmd:byte(idx)
		if chr == sep then
			for key, val in ipairs(amap) do
				dest[val], amap[key] = opts[argc], nil
			end
			argc = argc + 1
		else
			vname = memoryslots[string.char(chr)]
			if vname == nil then error(modulename ..
				', ‘setting’: Unknown slot ‘' ..
				string.char(chr) .. '’', 0) end
			table.insert(amap, vname)
		end
	end
	for key, val in ipairs(amap) do dest[val] = opts[argc] end
	return argc
end


-- Add a new stack of parameters to `ctx.children`
local function new_substack (ctx)
	local currsnap, newparams = ctx.n_children + 1, {}
	if ctx.children == nil then ctx.children = { newparams }
	else ctx.children[currsnap] = newparams end
	ctx.n_children = currsnap
	return newparams
end


-- Parse a raw argument containing a `sortfunctions` directive, or
-- `'without_sorting'`, or `nil`
local function load_sort_opt (raw_arg)
	if raw_arg == nil then return nil, 1, false end
	local trarg = raw_arg:match'^%s*(.-)%s*$'
	if trarg == 'without_sorting' then return nil, 2, false, trarg end
	local tmp = sortfunctions[trarg]
	if tmp == nil then return nil, 1, false, trarg end
	return tmp or nil, 2, true, trarg
end


-- Parse optional user arguments of type `...|[let/use]|[...]|[let/use]|[...]|
-- [number of additional parameters]|[parameter 1]|[parameter 2]|[...]`
local function load_child_opts (src, start_from, append_after, params)
	local tnamed, tmp1, tmp2
	local pin, tbl, mem = start_from, {}, {}
	while src[pin] ~= nil and src[pin + 1] ~= nil do
		tmp1 = src[pin]:match'^%s*(.*%S)'
		if tmp1 == 'let' and src[pin + 2] ~= nil then
			tmp1 = get_parameter_name(src[pin + 1])
			mem[tmp1], tbl[tmp1], pin = nil, src[pin + 2], pin + 3
--[[
		elseif tmp1 == 'expose' then
			tmp1 = get_parameter_name(src[pin + 1])
			tmp2 = params[tmp1]
			mem[tmp1], tbl[tmp1], pin = tmp2, tmp2, pin + 2
]]--
		elseif tmp1 == 'use' and src[pin + 2] ~= nil then
			tmp1 = get_parameter_name(src[pin + 2])
			tmp2 = params[tmp1]
			mem[tmp1], tbl[get_parameter_name(src[pin + 1])], pin =
				tmp2, tmp2, pin + 3
		else break end
	end
	local tnew = copy_or_ref_table(params, next(mem) == nil)
	for key in pairs(mem) do tnew[key] = nil end
	if pin ~= start_from then tnamed, tbl = tbl, {} end
	tmp1 = tonumber(src[pin])
	if tmp1 ~= nil and math.floor(tmp1) == tmp1 then
		if tmp1 &lt; 0 then tmp1 = -1 end
		tmp2 = append_after - pin
		for idx = pin + 1, pin + tmp1 do tbl[idx + tmp2] = src[idx] end
		pin = pin + tmp1 + 1
	end
	if tnamed ~= nil then
		for key, val in pairs(tnamed) do tbl[key] = val end
	end
	return tbl, pin, tnew, mem
end


-- Load the optional arguments of some of the `mapping_*` and `renaming_*`
-- class of modifiers
local function load_callback_opts (src, n_skip, default_style, params)
	local style, shf
	local tmp = src[n_skip + 1]
	if tmp ~= nil then style = mapping_styles[tmp:match'^%s*(.-)%s*$'] end
	if style == nil then style, shf = default_style, n_skip - 1
	else shf = n_skip end
	local n_exist, karg, varg = style[3], style[4], style[5]
	tmp = style[6]
	if tmp &gt; -1 then
		karg = src[tmp + shf]:match'^%s*(.-)%s*$'
		if karg == '0' or karg:find'^%-?[1-9]%d*$' ~= nil then
			karg = tonumber(karg)
			n_exist = math.max(n_exist, karg)
		end
	end
	tmp = style[7]
	if tmp &gt; -1 then
		varg = src[tmp + shf]:match'^%s*(.-)%s*$'
		if varg == '0' or varg:find'^%-?[1-9]%d*$' ~= nil then
			varg = tonumber(varg)
			n_exist = math.max(n_exist, varg)
		end
	end
	local dest, argc, tnew, mem = load_child_opts(src, style[2] + shf,
		n_exist, params)
	tmp = style[1]
	if (tmp == 3 or tmp == 2) and dest[karg] ~= nil then
		tmp = tmp - 2 end
	if (tmp == 3 or tmp == 1) and dest[varg] ~= nil then
		tmp = tmp - 1 end
	return dest, argc, tmp, karg, varg, tnew, mem
end


-- Parse the arguments of some of the `mapping_*` and `renaming_*` class of
-- modifiers
local function load_replace_args (opts, whoami)
	if opts[1] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No pattern string was given', 0) end
	if opts[2] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No replacement string was given', 0) end
	local ptn, repl, nmax, argc = opts[1], opts[2], tonumber(opts[3]), 3
	if nmax ~= nil or (opts[3] or ''):match'^%s*$' ~= nil then argc = 4 end
	local flg = opts[argc]
	if flg ~= nil then flg = mkeywords[flg:match'^%s*(.-)%s*$'] end
	if flg == 0 then flg = nil elseif flg ~= nil then argc = argc + 1 end
	return ptn, repl, nmax, flg, argc, (nmax ~= nil and nmax &lt; 1) or
		(flg == 3 and ptn == repl)
end


-- Parse the arguments of the `with_*_matching` class of modifiers
local function load_pattern_args (opts, whoami)
	local keyw
	local ptns, state, nptns, cnt = {}, 0, 0, 1
	for _, val in ipairs(opts) do
		if state == 0 then
			nptns, state = nptns + 1, -1
			ptns[nptns] = { val, false, false }
		else
			keyw = val:match'^%s*(.*%S)'
			if keyw == nil or mkeywords[keyw] == nil or (
				state &gt; 0 and mkeywords[keyw] &gt; 0
			) then break
			else
				state = mkeywords[keyw]
				if state &gt; 1 then ptns[nptns][2] = true end
				if state == 3 then ptns[nptns][3] = true end
			end
		end
		cnt = cnt + 1
	end
	if state == 0 then error(modulename .. ', ‘' .. whoami ..
		'’: No pattern was given', 0) end
	return ptns, nptns, cnt
end


-- Load the optional arguments of the `parsing`, `reinterpreting` and
-- `evaluating` modifiers
local function load_parse_opts (opts, start_from, isp, psp)
	local tmp
	local optslots, noptslots, argc, trimn, trimu, iplain, pplain =
		{ true, true, true }, 3, start_from, true, false, true, true
	repeat
		noptslots, tmp = noptslots - 1, opts[argc]
		if tmp == nil then break end
		tmp = tmp:match'^%s*(.-)%s*$'
		if optslots[1] ~= nil and trim_parse_opts[tmp] ~= nil then
			tmp = trim_parse_opts[tmp]
			trimn, trimu, optslots[1] = tmp[1], tmp[2], nil
		elseif optslots[2] ~= nil and isep_parse_opts[tmp] ~= nil then
			argc = argc + 1
			iplain, isp, optslots[2] = isep_parse_opts[tmp],
				opts[argc], nil
		elseif optslots[3] ~= nil and psep_parse_opts[tmp] ~= nil then
			argc = argc + 1
			pplain, psp, optslots[3] = psep_parse_opts[tmp],
				opts[argc], nil
		else break end
		argc = argc + 1
	until noptslots &lt; 1
	return isp, iplain, psp, pplain, trimn, trimu, argc
end


-- Map parameters' values using a custom callback and a referenced table
local value_maps = {
	[0] = function (tbl, margs, karg, varg, fn)
		for key in pairs(tbl) do tbl[key] = fn() end
	end,
	[1] = function (tbl, margs, karg, varg, fn)
		for key, val in pairs(tbl) do
			margs[varg] = val
			tbl[key] = fn()
		end
	end,
	[2] = function (tbl, margs, karg, varg, fn)
		for key in pairs(tbl) do
			margs[karg] = key
			tbl[key] = fn()
		end
	end,
	[3] = function (tbl, margs, karg, varg, fn)
		for key, val in pairs(tbl) do
			margs[karg], margs[varg] = key, val
			tbl[key] = fn()
		end
	end
}


-- Private table for `map_names()`
local name_thieves = {
	[0] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[1] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[varg] = val
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[2] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[karg] = key
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[3] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[karg], rargs[varg] = key, val
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end
}


-- Map parameters' names using a custom callback and a referenced table
local function map_names (tbl, rargs, karg, varg, looptype, fn)
	local cache = {}
	name_thieves[looptype](cache, tbl, rargs, karg, varg, fn)
	for key, val in pairs(cache) do tbl[key] = val end
end


-- Return a new table that contains `src` regrouped according to the numeric
-- suffixes in its keys
local function make_groups (src)
	-- NOTE: `src` might be the original metatable!
	local prefix, gid
	local groups = {}
	for key, val in pairs(src) do
		-- `key` must only be a string or a number...
		if type(key) == 'string' then
			prefix, gid = key:match'^%s*(.-)%s*(%-?%d*)%s*$'
			gid = tonumber(gid) or ''
		else
			prefix, gid = '', key
		end
		if groups[gid] == nil then groups[gid] = {} end
		if prefix == '0' or prefix:find'^%-?[1-9]%d*$' ~= nil then
			prefix = tonumber(prefix)
			if prefix &lt; 1 then prefix = prefix - 1 end
		end
		groups[gid][prefix] = val
	end
	return groups
end


-- Split into parts a string containing the `$#` and `$@` placeholders and
-- return the information as a skeleton table, a canvas table and a length
local function parse_placeholder_string (target)
	local idx, s_pos, skel, canvas = 1, 1, {}, {}
	local e_pos = string.find(target, '%$[@#]', 1, false)
	while e_pos ~= nil do
		canvas[idx] = target:sub(s_pos, e_pos - 1)
		skel[idx + 1] = target:sub(e_pos, e_pos + 1) == '$@'
		idx = idx + 2
		s_pos = e_pos + 2
		e_pos = string.find(target, '%$[@#]', s_pos, false)
	end
	if (s_pos &gt; target:len()) then idx = idx - 1
	else canvas[idx] = target:sub(s_pos) end
	return skel, canvas, idx
end


-- Populate a table by parsing a parameter string (heavy lifting for `parsing`,
-- `reinterpreting` and `evaluating`)
local function parse_parameter_string (tbl, str, isp, ipl, psp, ppl, trn, tru)
	local key, val, spos1, spos2, pos1, pos2
	local pos3, idx, lenplone = 0, 1, #str + 1
	if isp == nil or isp == '' then
		if psp == nil or psp == '' then
			if tru then tbl[idx] = str:match'^%s*(.-)%s*$'
			else tbl[idx] = str end
			return idx
		end
		spos1, spos2 = str:find(psp, 1, ppl)
		if spos1 == nil then
			key = idx
			if tru then val = str:match'^%s*(.-)%s*$'
			else val = str end
			idx = idx + 1
		else
			key = get_parameter_name(str:sub(1, spos1 - 1))
			val = str:sub(spos2 + 1)
			if trn then val = val:match'^%s*(.-)%s*$' end
		end
		tbl[key] = val
		return idx
	end
	if psp == nil or psp == '' then
		repeat
			pos1 = pos3 + 1
			pos2, pos3 = str:find(isp, pos1, ipl)
			val = str:sub(pos1, (pos2 or lenplone) - 1)
			if tru then val = val:match'^%s*(.-)%s*$' end
			tbl[idx], idx = val, idx + 1
		until pos2 == nil
		return idx
	end
	repeat
		pos1 = pos3 + 1
		pos2, pos3 = str:find(isp, pos1, ipl)
		val = str:sub(pos1, (pos2 or lenplone) - 1)
		spos1, spos2 = val:find(psp, 1, ppl)
		if spos1 == nil then
			key = idx
			if tru then val = val:match'^%s*(.-)%s*$' end
			idx = idx + 1
		else
			key = get_parameter_name(val:sub(1, spos1 - 1))
			val = val:sub(spos2 + 1)
			if trn then val = val:match'^%s*(.-)%s*$' end
		end
		tbl[key] = val
	until pos2 == nil
	return idx
end


-- Heavy lifting for `snapshotting` and `remembering`
local function make_child (ctx, src, whoami)
	local len = tonumber(ctx.pipe[1])
	if len == nil or len == 0 then
		local stack = new_substack(ctx)
		for key, val in pairs(src) do stack[key] = val end
		return len == nil and 1 or 2
	end
	if len &lt; 0 or math.floor(len) ~= len then error(modulename ..
		', ‘' .. whoami .. '’: The number of parameters to copy must be an integer greater than zero', 0) end
	copy_table_maxn(new_substack(ctx), src, len)
	return 2
end


-- Heavy lifting for `setting_by_flushing`, `combining` and `combining_values`
local function set_strings_from_substack (ctx, dest, whoami)
	if ctx.n_children &lt; 1 then error(modulename ..
		', ‘' .. whoami .. '’: There are no substacks to flush', 0) end
	local currsnap = ctx.n_children
	local stack = ctx.children[currsnap]
	for key, val in pairs(memoryslots) do
		if stack[key] ~= nil then dest[val] = stack[key] end
	end
	ctx.children[currsnap], ctx.n_children = nil, currsnap - 1
end


-- Heavy lifting for `combining` and `combining_values`
local function combine_parameters (ctx, keyval_fn, whoami)
	-- NOTE: `ctx.params` might be the original metatable! This function
	-- MUST create a copy of it before returning
	local opts = ctx.pipe
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No parameter name was provided', 0) end
	local argc
	local tbl, vars = ctx.params, {}
	local sortfn, varsarg0, do_sort, tmp = load_sort_opt(opts[2])
	if varsarg0 == 2 then tmp = opts[3] and opts[3]:match'^%s*(.-)%s*$' end
	if tmp == 'with_flushed_glue' then
		varsarg0 = varsarg0 + 1
		argc = set_strings_from_opts(vars, opts, varsarg0 + 1)
		set_strings_from_substack(ctx, vars, whoami)
	else
		if tmp == 'without_flushed_glue' then varsarg0 = varsarg0 + 1 end
		argc = set_strings_from_opts(vars, opts, varsarg0 + 1)
	end
	if argc &lt; varsarg0 then error(modulename ..
		', ‘' .. whoami .. '’: No setting directive was given', 0) end
	if next(tbl) == nil then
		if vars.ifngiven ~= nil then ctx.params =
			{ [get_parameter_name(ctx.pipe[1])] = vars.ifngiven }
		elseif tbl == ctx.oparams then ctx.params = {} end
		return argc
	end
	local cache, len
	if do_sort then
		local words
		cache, words, len, tmp = get_key_list_sorted(tbl, sortfn)
		for idx = 1, tmp do cache[len + idx] = words[idx] end
		len = len + tmp
	else
		len, cache = 0, {}
		for key in pairs(tbl) do
			len = len + 1
			cache[len] = key
		end
	end
	local pmap, nss, kvs, pps = {}, 0, vars.pairsep or '', vars.itersep or ''
	for idx = 1, len do
		tmp, pmap[nss + 1] = cache[idx], pps
		pmap[nss + 2] = keyval_fn(tmp, tbl[tmp], kvs)
		nss = nss + 2
	end
	tmp = vars.oxfordsep or vars.lastsep
	if tmp ~= nil and nss &gt; 4 then pmap[nss - 1] = tmp
	elseif nss &gt; 2 and vars.lastsep ~= nil then
		pmap[nss - 1] = vars.lastsep
	end
	pmap[1] = vars.header or ''
	if vars.footer ~= nil then pmap[nss + 1] = vars.footer end
	ctx.params = { [get_parameter_name(ctx.pipe[1])] = table.concat(pmap) }
	return argc
end


-- Concatenate the numeric keys from the table of parameters to the numeric
-- keys from the table of options; non-numeric keys from the table of options
-- will prevail over colliding non-numeric keys from the table of parameters
local function concat_params (ctx)
	local retval, tbl, nmax = {}, ctx.params, table.maxn(ctx.pipe)
	if ctx.subset == 1 then
		-- We need only the sequence
		for key, val in ipairs(tbl) do retval[key + nmax] = val end
	else
		if ctx.subset == -1 then
			for key in ipairs(tbl) do tbl[key] = nil end
		end
		for key, val in pairs(tbl) do
			if type(key) == 'number' and key &gt; 0 then
				retval[key + nmax] = val
			else retval[key] = val end
		end
	end
	for key, val in pairs(ctx.pipe) do retval[key] = val end
	return retval
end


-- Flush the parameters by calling a custom function for each value (after this
-- function has been invoked `ctx.params` will be no longer usable)
local function flush_params (ctx, fn)
	local tbl = ctx.params
	if ctx.subset == 1 then
		for key, val in ipairs(tbl) do fn(key, val) end
		return
	end
	if ctx.subset == -1 then
		for key, val in ipairs(tbl) do tbl[key] = nil end
	end
	if ctx.sorttype &gt; 0 then
		local nums, words, nn, nw = get_key_list_sorted(tbl, natural_sort)
		if ctx.sorttype == 2 then
			for idx = 1, nw do fn(words[idx], tbl[words[idx]]) end
			for idx = 1, nn do fn(nums[idx], tbl[nums[idx]]) end
			return
		end
		for idx = 1, nn do fn(nums[idx], tbl[nums[idx]]) end
		for idx = 1, nw do fn(words[idx], tbl[words[idx]]) end
		return
	end
	if ctx.subset ~= -1 then
		for key, val in ipairs(tbl) do
			fn(key, val)
			tbl[key] = nil
		end
	end
	for key, val in pairs(tbl) do fn(key, val) end
end


-- Flush the parameters by calling one of two custom functions for each value
-- (after this function has been invoked `ctx.params` will be no longer usable)
local function mixed_flush_params (ctx, fn_seq, fn_oth)
	if ctx.subset == 1 then
		for key, val in ipairs(ctx.params) do fn_seq(key, val) end
		return
	end
	if ctx.subset == -1 then
		flush_params(ctx, fn_oth)
		return
	end
	local tbl = ctx.params
	if ctx.sorttype &gt; 0 then
		local nums, words, nn, nw = get_key_list_sorted(tbl, natural_sort)
		local sequence = {}
		for key, val in ipairs(tbl) do sequence[key] = val end
		if ctx.sorttype == 2 then
			for idx = 1, nw do fn_oth(words[idx], tbl[words[idx]]) end
		end
		for idx = 1, nn do
			if sequence[nums[idx]] then fn_seq(nums[idx], sequence[nums[idx]])
			else fn_oth(nums[idx], tbl[nums[idx]]) end
		end
		if ctx.sorttype ~= 2 then
			for idx = 1, nw do fn_oth(words[idx], tbl[words[idx]]) end
		end
		return
	end
	for key, val in ipairs(tbl) do
		fn_seq(key, val)
		tbl[key] = nil
	end
	for key, val in pairs(tbl) do fn_oth(key, val) end
end


-- Finalize and return a concatenated list
local function finalize_and_return_concatenated_list (ctx, lst, len, modsize)
	if len &gt; 0 then
		local tmp = ctx.oxfordsep or ctx.lastsep
		if tmp ~= nil and len &gt; modsize * 2 then
			lst[len - modsize + 1] = tmp
		elseif len &gt; modsize and ctx.lastsep ~= nil then
			lst[len - modsize + 1] = ctx.lastsep
		end
		lst[1] = ctx.header or ''
		if ctx.footer ~= nil then lst[len + 1] = ctx.footer end
		ctx.text = table.concat(lst)
	else ctx.text = ctx.ifngiven or '' end
end


	---                                        ---
	---     PUBLIC ENVIRONMENT                 ---
	---    ________________________________    ---
	---                                        ---


	--[[ Modifiers ]]--
	-------------------


-- Syntax:  #invoke:params|sequential|pipe to
library.sequential = function (ctx)
	if ctx.subset == 1 then error(modulename ..
		': The ‘sequential’ directive has been provided more than once', 0) end
	if ctx.subset == -1 then error(modulename ..
		': The two directives ‘non-sequential’ and ‘sequential’ are in contradiction with each other', 0) end
	if ctx.sorttype &gt; 0 then error(modulename ..
		': The ‘all_sorted’ and ‘reassorted’ directives are redundant when followed by ‘sequential’', 0) end
	ctx.iterfunc, ctx.subset = ipairs, 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|non-sequential|pipe to
library['non-sequential'] = function (ctx)
	if ctx.subset == -1 then error(modulename ..
		': The ‘non-sequential’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The two directives ‘sequential’ and ‘non-sequential’ are in contradiction with each other', 0) end
	ctx.iterfunc, ctx.subset = pairs, -1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|all_sorted|pipe to
library.all_sorted = function (ctx)
	if ctx.sorttype == 1 then error(modulename ..
		': The ‘all_sorted’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The ‘all_sorted’ directive is redundant after ‘sequential’', 0) end
	if ctx.sorttype == 2 then error(modulename ..
		': The two directives ‘reassorted’ and ‘sequential’ are in contradiction with each other', 0) end
	ctx.sorttype = 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|reassorted|pipe to
library.reassorted = function (ctx)
	if ctx.sorttype == 2 then error(modulename ..
		': The ‘reassorted’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The ‘reassorted’ directive is redundant after ‘sequential’', 0) end
	if ctx.sorttype == 1 then error(modulename ..
		': The two directives ‘sequential’ and ‘reassorted’ are in contradiction with each other', 0) end
	ctx.sorttype = 2
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|setting|directives|...|pipe to
library.setting = function (ctx)
	local argc = set_strings_from_opts(ctx, ctx.pipe, 1)
	if argc &lt; 2 then error(modulename ..
		', ‘setting’: No directive was given', 0) end
	return context_iterate(ctx, argc + 1)
end


-- Syntax:  #invoke:params|scoring|new parameter name|[container]|pipe to
library.scoring = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘scoring’: No parameter name was provided', 0) end
	local tmp
	local retval, opts = 0, ctx.pipe
	for _ in pairs(ctx.params) do retval = retval + 1 end
	if opts[2] ~= nil then tmp = opts[2]:match'^%s*(.*%S)' end
	if tmp == 'in_substack' then
		new_substack(ctx)[get_parameter_name(opts[1])] = tostring(retval)
		return context_iterate(ctx, 3)
	end
	ctx.params[get_parameter_name(opts[1])] = tostring(retval)
	return context_iterate(ctx, tmp == 'here' and 3 or 2)
end


-- Syntax:  #invoke:params|squeezing|pipe to
library.squeezing = function (ctx)
	local store, indices, tbl, newlen = {}, {}, ctx.params, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			newlen = newlen + 1
			indices[newlen], store[key], tbl[key] = key, val, nil
		end
	end
	table.sort(indices)
	for idx = 1, newlen do tbl[idx] = store[indices[idx]] end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|filling_the_gaps|pipe to
library.filling_the_gaps = function (ctx)
	local newval, tbl, tmp, nmin, nmax, nnums =
		ctx.pipe[1], ctx.params, {}, 1, nil, -1
	if newval == nil then error(modulename ..
		', ‘filling_the_gaps’: No value was provided', 0) end
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			if nmax == nil then
				if key &lt; nmin then nmin = key end
				nmax = key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
			tmp[key], nnums = val, nnums + 1
		end
	end
	if nmax ~= nil and nmax - nmin &gt; nnums then
		ctx.n_available = ctx.n_available + nmin + nnums - nmax
		if ctx.n_available &lt; 0 then error(modulename ..
			', ‘filling_the_gaps’: It is possible to fill at most ' ..
			tostring(maxfill) .. ' parameters', 0) end
		for idx = nmin, nmax, 1 do tbl[idx] = newval end
		for key, val in pairs(tmp) do tbl[key] = val end
	end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|clearing|pipe to
library.clearing = function (ctx)
	local tbl, numerics = ctx.params, {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key], tbl[key] = val, nil
		end
	end
	for key, val in ipairs(numerics) do tbl[key] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|cutting|left cut|right cut|pipe to
library.cutting = function (ctx)
	local lcut = tonumber(ctx.pipe[1])
	if lcut == nil or math.floor(lcut) ~= lcut then error(modulename ..
		', ‘cutting’: Left cut must be an integer number', 0) end
	local rcut = tonumber(ctx.pipe[2])
	if rcut == nil or math.floor(rcut) ~= rcut then error(modulename ..
		', ‘cutting’: Right cut must be an integer number', 0) end
	local tbl = ctx.params
	local len = #tbl
	if lcut &lt; 0 then lcut = len + lcut end
	if rcut &lt; 0 then rcut = len + rcut end
	local tot = lcut + rcut
	if tot &gt; 0 then
		local cache = {}
		if tot &gt;= len then
			for key in ipairs(tbl) do tbl[key] = nil end
			tot = len
		else
			for idx = len - rcut + 1, len, 1 do tbl[idx] = nil end
			for idx = 1, lcut, 1 do tbl[idx] = nil end
		end
		for key, val in pairs(tbl) do
			if type(key) == 'number' and key &gt; 0 then
				if key &gt; len then cache[key - tot] = val
				else cache[key - lcut] = val end
				tbl[key] = nil
			end
		end
		for key, val in pairs(cache) do tbl[key] = val end
	end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|cropping|left crop|right crop|pipe to
library.cropping = function (ctx)
	local lcut = tonumber(ctx.pipe[1])
	if lcut == nil or math.floor(lcut) ~= lcut then error(modulename ..
		', ‘cropping’: Left crop must be an integer number', 0) end
	local rcut = tonumber(ctx.pipe[2])
	if rcut == nil or math.floor(rcut) ~= rcut then error(modulename ..
		', ‘cropping’: Right crop must be an integer number', 0) end
	local tbl = ctx.params
	local nmin, nmax
	for key in pairs(tbl) do
		if type(key) == 'number' then
			if nmin == nil then nmin, nmax = key, key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
		end
	end
	if nmin ~= nil then
		local len = nmax - nmin + 1
		if lcut &lt; 0 then lcut = len + lcut end
		if rcut &lt; 0 then rcut = len + rcut end
		if lcut + rcut - len &gt; -1 then
			for key in pairs(tbl) do
				if type(key) == 'number' then tbl[key] = nil end
			end
		elseif lcut + rcut &gt; 0 then
			for idx = nmax - rcut + 1, nmax do tbl[idx] = nil end
			for idx = nmin, nmin + lcut - 1 do tbl[idx] = nil end
			local lshift = nmin + lcut - 1
			if lshift &gt; 0 then
				for idx = lshift + 1, nmax, 1 do
					tbl[idx - lshift], tbl[idx] =
						tbl[idx], nil
				end
			end
		end
	end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|purging|start offset|length|pipe to
library.purging = function (ctx)
	local idx = tonumber(ctx.pipe[1])
	if idx == nil or math.floor(idx) ~= idx then error(modulename ..
		', ‘purging’: Start offset must be an integer number', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘purging’: Length must be an integer number', 0) end
	local tbl = ctx.params
	if len &lt; 1 then
		len = len + table.maxn(tbl)
		if idx &gt; len then return context_iterate(ctx, 3) end
		len = len - idx + 1
	end
	ctx.params = copy_table_reduced(tbl, idx, len)
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|backpurging|start offset|length|pipe to
library.backpurging = function (ctx)
	local last = tonumber(ctx.pipe[1])
	if last == nil or math.floor(last) ~= last then error(modulename ..
		', ‘backpurging’: Start offset must be an integer number', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘backpurging’: Length must be an integer number', 0) end
	local idx
	local tbl = ctx.params
	if len &gt; 0 then
		idx = last - len + 1
	else
		for key in pairs(tbl) do
			if type(key) == 'number' and (idx == nil or
				key &lt; idx) then idx = key end
		end
		if idx == nil then return context_iterate(ctx, 3) end
		idx = idx - len
		if last &lt; idx then return context_iterate(ctx, 3) end
		len = last - idx + 1
	end
	ctx.params = copy_table_reduced(ctx.params, idx, len)
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|shifting|addend|pipe to
library.shifting = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local nshift = tonumber(ctx.pipe[1])
	if nshift == nil or nshift == 0 or math.floor(nshift) ~= nshift then
		error(modulename .. ', ‘shifting’: A non-zero integer number must be provided', 0) end
	local tbl = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'number' then tbl[key + nshift] = val
		else tbl[key] = val end
	end
	ctx.params = tbl
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|reversing_numeric_names|pipe to
library.reversing_numeric_names = function (ctx)
	local tbl, numerics, nmax = ctx.params, {}, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key], tbl[key] = val, nil
			if key &gt; nmax then nmax = key end
		end
	end
	for key, val in pairs(numerics) do tbl[nmax - key + 1] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|pivoting_numeric_names|pipe to
--[[
library.pivoting_numeric_names = function (ctx)
	local tbl = ctx.params
	local shift = #tbl + 1
	if shift &lt; 2 then return library.reversing_numeric_names(ctx) end
	local numerics = {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key] = val
			tbl[key] = nil
		end
	end
	for key, val in pairs(numerics) do tbl[shift - key] = val end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|mirroring_numeric_names|pipe to
--[[
library.mirroring_numeric_names = function (ctx)
	local nmax, nmin
	local tbl, numerics = ctx.params, {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key] = val
			tbl[key] = nil
			if nmax == nil then nmin, nmax = key, key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
		end
	end
	for key, val in pairs(numerics) do tbl[nmax + nmin - key] = val end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|swapping_numeric_names|pipe to
--[[
library.swapping_numeric_names = function (ctx)
	local tmp
	local tbl, cache, nsize = ctx.params, {}, 0
	for key in pairs(tbl) do
		if type(key) == 'number' then
			nsize = nsize + 1
			cache[nsize] = key
		end
	end
	table.sort(cache)
	for idx = math.floor(nsize / 2), 1, -1 do
		tmp = tbl[cache[idx] ]
		tbl[cache[idx] ] = tbl[cache[nsize - idx + 1] ]
		tbl[cache[nsize - idx + 1] ] = tmp
	end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|sorting_sequential_values|[criterion]|pipe to
library.sorting_sequential_values = function (ctx)
	local sortfn
	if ctx.pipe[1] ~= nil then
		sortfn = sortfunctions[ctx.pipe[1]:match'^%s*(.-)%s*$']
	end
	if sortfn then table.sort(ctx.params, sortfn)
	else table.sort(ctx.params) end -- i.e. either `false` or `nil`
	if sortfn == nil then return context_iterate(ctx, 1) end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|splicing|[add to position]|position|increment|
--            [number of elements to write]|...|pipe to
library.splicing = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp2, argc, pos, refp
	local opts, tbl = ctx.pipe, ctx.params
	local tmp1 = opts[1]
	if tmp1 ~= nil then
		tmp2 = tonumber(tmp1)
		if tmp2 == nil or math.floor(tmp2) ~= tmp2 then
			pos, argc, tmp2 = tonumber(opts[2]), 4,
				tmp1:match'^%s*(.*%S)'
			if tmp2 ~= nil then
				refp = position_references[tmp2]
				if refp == nil then error(modulename ..
					', ‘splicing’: ‘' .. tostring(tmp2) ..
					'’ is not a valid first argument', 0) end
			else refp = 0 end
		else pos, argc, refp = tmp2, 3, 0 end
	else pos, argc, refp = tonumber(opts[2]), 4, 0 end
	if pos == nil or math.floor(pos) ~= pos then error(modulename ..
		', ‘splicing’: The position must be an integer number', 0) end
	local len = tonumber(opts[argc - 1])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘splicing’: The increment must be an integer number', 0) end
	if refp == 2 then
		for _ in ipairs(tbl) do pos = pos + 1 end
		refp = 0
	end
	tmp1, tmp2 = nil, nil
	if refp ~= 0 or len ~= 0 then
		for key, val in pairs(tbl) do
			if type(key) == 'number' then
				if tmp1 == nil then tmp1, tmp2 = key, key
				elseif key &lt; tmp1 then tmp1 = key
				elseif key &gt; tmp2 then tmp2 = key end
			end
		end
	end
	if tmp2 == nil then len = 0
	elseif refp == 3 then pos = pos + tmp2
	elseif refp == 1 then pos = pos + tmp1 end
	if len &gt; 0 and pos + len &gt; tmp1 and pos &lt;= tmp2 then
		tbl = copy_table_expanded(tbl, pos, len)
	elseif len &lt; 0 and pos - len &gt; tmp1 and pos &lt;= tmp2 then
		tbl = copy_table_reduced(tbl, pos, -len)
	else tbl = copy_or_ref_table(tbl, tbl ~= ctx.oparams) end
	ctx.params = tbl
	tmp1 = tonumber(opts[argc])
	if len == 0 and (tmp1 == nil or tmp1 &lt; 1) then error(modulename ..
		', ‘splicing’: When the increment is zero the number of elements to add cannot be zero', 0) end
	if tmp1 == nil or tmp1 &lt; 0 or math.floor(tmp1) ~= tmp1 then
		return context_iterate(ctx, argc)
	end
	tmp2 = argc - pos + 1
	for key = pos, pos + tmp1 - 1 do tbl[key] = opts[key + tmp2] end
	return context_iterate(ctx, argc + tmp1 + 1)
end


-- Syntax:  #invoke:params|imposing|name|value|pipe to
library.imposing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘imposing’: Missing parameter name to impose', 0) end
	ctx.params[get_parameter_name(ctx.pipe[1])] = ctx.pipe[2]
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|providing|name|value|pipe to
library.providing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘providing’: Missing parameter name to provide', 0) end
	local key = get_parameter_name(ctx.pipe[1])
	if ctx.params[key] == nil then ctx.params[key] = ctx.pipe[2] end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|reassigning|source|destination|[mode]|pipe to
library.reassigning = function (ctx)
	local opts, tbl = ctx.pipe, ctx.params
	if opts[1] == nil then error(modulename ..
		', ‘reassigning’: Missing source parameter', 0) end
	if opts[2] == nil then error(modulename ..
		', ‘reassigning’: Missing destination parameter', 0) end
	local mode, argc
	local src = get_parameter_name(opts[1])
	local val = tbl[src]
	if opts[3] ~= nil then mode = a_modes[opts[3]:match'^%s*(.-)%s*$'] end
	if mode == nil then mode, argc = 0, 3 else argc = 4 end
	if val == nil and mode &gt; 1 then return context_iterate(ctx, argc) end
	local dest = get_parameter_name(opts[2])
	local tmp = tbl[dest] == nil
	if mode % 2 == 0 or (mode == 7 and tmp) then tbl[src] = nil end
	if tmp or mode &lt; 4 then tbl[dest] = val end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|discarding|name|[how many]|pipe to
library.discarding = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘discarding’: Missing parameter name to discard', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil then
		ctx.params[get_parameter_name(ctx.pipe[1])] = nil
		return context_iterate(ctx, 2)
	end
	local key = tonumber(ctx.pipe[1])
	if key == nil or math.floor(key) ~= key then error(modulename ..
		', ‘discarding’: A range was provided, but the initial parameter name is not an integer number', 0) end
	if len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘discarding’: A range can only be an integer number greater than zero', 0) end
	for idx = key, key + len - 1 do ctx.params[idx] = nil end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|excluding_non-numeric_names|pipe to
library['excluding_non-numeric_names'] = function (ctx)
	local tmp = ctx.params
	for key, val in pairs(tmp) do
		if type(key) ~= 'number' then tmp[key] = nil end
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|excluding_numeric_names|pipe to
library.excluding_numeric_names = function (ctx)
	local tmp = ctx.params
	for key, val in pairs(tmp) do
		if type(key) == 'number' then tmp[key] = nil end
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|with_name_matching|target 1|[plain flag 1]|[or]
--            |[target 2]|[plain flag 2]|[or]|[...]|[target N]|[plain flag
--            N]|pipe to
library.with_name_matching = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp, ptn
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_name_matching')
	local tbl, newparams = ctx.params, {}
	for idx = 1, nptns do
		ptn = targets[idx]
		if ptn[3] then
			tmp = ptn[1]
			if tmp == '0' or tmp:find'^%-?[1-9]%d*$' ~= nil then
				tmp = tonumber(tmp)
			end
			newparams[tmp] = tbl[tmp]
		else
			for key, val in pairs(tbl) do
				if tostring(key):find(ptn[1], 1, ptn[2]) then
					newparams[key] = val
				end
			end
		end
	end
	ctx.params = newparams
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_name_not_matching|target 1|[plain flag 1]
--            |[and]|[target 2]|[plain flag 2]|[and]|[...]|[target N]|[plain
--            flag N]|pipe to
library.with_name_not_matching = function (ctx)
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_name_not_matching')
	local tbl = ctx.params
	if nptns == 1 and targets[1][3] then
		local tmp = targets[1][1]
		if tmp == '0' or tmp:find'^%-?[1-9]%d*$' ~= nil then
			tbl[tonumber(tmp)] = nil
		else tbl[tmp] = nil end
		return context_iterate(ctx, argc)
	end
	local yesmatch, ptn
	for key in pairs(tbl) do
		yesmatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if tostring(key) ~= ptn[1] then
					yesmatch = false
					break
				end
			elseif not tostring(key):find(ptn[1], 1, ptn[2]) then
				yesmatch = false
				break
			end
		end
		if yesmatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_value_matching|target 1|[plain flag 1]|[or]
--            |[target 2]|[plain flag 2]|[or]|[...]|[target N]|[plain flag
--            N]|pipe to
library.with_value_matching = function (ctx)
	local nomatch, ptn
	local tbl = ctx.params
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_value_matching')
	for key, val in pairs(tbl) do
		nomatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if val == ptn[1] then
					nomatch = false
					break
				end
			elseif val:find(ptn[1], 1, ptn[2]) then
				nomatch = false
				break
			end
		end
		if nomatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_value_not_matching|target 1|[plain flag 1]
--            |[and]|[target 2]|[plain flag 2]|[and]|[...]|[target N]|[plain
--            flag N]|pipe to
library.with_value_not_matching = function (ctx)
	local yesmatch, ptn
	local tbl = ctx.params
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_value_not_matching')
	for key, val in pairs(tbl) do
		yesmatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if val ~= ptn[1] then
					yesmatch = false
					break
				end
			elseif not val:find(ptn[1], 1, ptn[2]) then
				yesmatch = false
				break
			end
		end
		if yesmatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|keeping_at_most|number of parameters to pick|pipe to
library.keeping_at_most = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local len = tonumber(ctx.pipe[1])
	if len == nil or len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘keeping_at_most’: The number of parameters to keep must be an integer greater than zero', 0) end
	ctx.params = copy_table_maxn({}, ctx.params, len)
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|trimming_values|pipe to
library.trimming_values = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:match'^%s*(.-)%s*$' end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_to_lowercase|pipe to
library.mapping_to_lowercase = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:lower() end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_to_uppercase|pipe to
library.mapping_to_uppercase = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:upper() end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_by_calling|template name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            parameters]|[parameter 1]|[parameter 2]|[...]|[parameter N]|pipe to
library.mapping_by_calling = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘mapping_by_calling’: No template name was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.values_only, ctx.params)
	local model = { title = tname, args = margs }
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return ctx.frame:expandTemplate(model)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_invoking|module name|function name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.mapping_by_invoking = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘mapping_by_invoking’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘mapping_by_invoking’: No function name was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 2, mapping_styles.values_only, ctx.params)
	local model = { title = 'Module:' .. mname, args = margs }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘mapping_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return tostring(mfunc(ctx.frame:newChild(model)))
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_magic|parser function|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.mapping_by_magic = function (ctx)
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘mapping_by_magic’: No parser function was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.values_only, ctx.params)
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return ctx.frame:callParserFunction(magic, margs)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_replacing|target|replace|[count]|[plain
--            flag]|pipe to
library.mapping_by_replacing = function (ctx)
	local ptn, repl, nmax, flg, argc, die =
		load_replace_args(ctx.pipe, 'mapping_by_replacing')
	if die then return context_iterate(ctx, argc) end
	local tbl = ctx.params
	if flg == 3 then
		for key, val in pairs(tbl) do
			if val == ptn then tbl[key] = repl end
		end
	else
		if flg == 2 then
			-- Copied from Module:String's `str._escapePattern()`
			ptn = ptn:gsub('[%(%)%.%%%+%-%*%?%[%^%$%]]', '%%%0')
		end
		for key, val in pairs(tbl) do
			tbl[key] = val:gsub(ptn, repl, nmax)
		end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_mixing|mixing string|pipe to
library.mapping_by_mixing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘mapping_by_mixing’: No mixing string was provided', 0) end
	local tbl, mix = ctx.params, ctx.pipe[1]
	if mix == '$#' then
		for key in pairs(tbl) do tbl[key] = tostring(key) end
		return context_iterate(ctx, 2)
	end
	local skel, cnv, n_parts = parse_placeholder_string(mix)
	for key, val in pairs(tbl) do
		for idx = 2, n_parts, 2 do
			if skel[idx] then cnv[idx] = val
			else cnv[idx] = tostring(key) end
		end
		tbl[key] = table.concat(cnv)
	end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|mapping_to_names|pipe to
--[[
library.mapping_to_names = function (ctx)
	local tbl = ctx.params
	for key in pairs(tbl) do tbl[key] = tostring(key) end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|renaming_to_lowercase|pipe to
library.renaming_to_lowercase = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'string' then cache[key:lower()] = val else
		cache[key] = val end
	end
	ctx.params = cache
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|renaming_to_uppercase|pipe to
library.renaming_to_uppercase = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'string' then cache[key:upper()] = val else
		cache[key] = val end
	end
	ctx.params = cache
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|renaming_to_sequence|[sort order]|pipe to
library.renaming_to_sequence = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache, len
	local tbl = ctx.params
	local sortfn, argc, do_sort = load_sort_opt(ctx.pipe[1])
	if do_sort then
		local words, wl
		cache, words, len, wl = get_key_list_sorted(tbl, sortfn)
		for idx = 1, len do cache[idx] = tbl[cache[idx]] end
		for idx = 1, wl do cache[len + idx] = tbl[words[idx]] end
	else
		len, cache = 0, {}
		for _, val in pairs(tbl) do
			len = len + 1
			cache[len] = val
		end
	end
	ctx.params = cache
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_calling|template name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            parameters]|[parameter 1]|[parameter 2]|[...]|[parameter N]|pipe to
library.renaming_by_calling = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘renaming_by_calling’: No template name was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.names_only, ctx.params)
	local model = { title = tname, args = rargs }
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return ctx.frame:expandTemplate(model)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_invoking|module name|function
--            name|[call style]|[let/use]|[...]|[let/use]|[...]|[number of
--            additional arguments]|[argument 1]|[argument 2]|[...]|[argument
--            N]|pipe to
library.renaming_by_invoking = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘renaming_by_invoking’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘renaming_by_invoking’: No function name was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 2, mapping_styles.names_only, ctx.params)
	local model = { title = 'Module:' .. mname, args = rargs }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘renaming_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return tostring(mfunc(ctx.frame:newChild(model)))
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_magic|parser function|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.renaming_by_magic = function (ctx)
	local opts = ctx.pipe
	local magic
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘renaming_by_magic’: No parser function was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.names_only, ctx.params)
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return ctx.frame:callParserFunction(magic, rargs)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_replacing|target|replace|[count]|[plain
--            flag]|pipe to
library.renaming_by_replacing = function (ctx)
	local ptn, repl, nmax, flg, argc, die =
		load_replace_args(ctx.pipe, 'renaming_by_replacing')
	if die then return context_iterate(ctx, argc) end
	local tbl = ctx.params
	if flg == 3 then
		ptn = get_parameter_name(ptn)
		local val = tbl[ptn]
		if val ~= nil then
			tbl[ptn], tbl[get_parameter_name(repl)] = nil, val
		end
	else
		if flg == 2 then
			-- Copied from Module:String's `str._escapePattern()`
			ptn = ptn:gsub('[%(%)%.%%%+%-%*%?%[%^%$%]]', '%%%0')
		end
		local cache = {}
		for key, val in pairs(tbl) do
			steal_if_renamed(val, tbl, key, cache,
				tostring(key):gsub(ptn, repl, nmax))
		end
		for key, val in pairs(cache) do tbl[key] = val end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_mixing|mixing string|pipe to
library.renaming_by_mixing = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘renaming_by_mixing’: No mixing string was provided', 0) end
	local mix = ctx.pipe[1]:match'^%s*(.-)%s*$'
	local cache = {}
	if mix == '$@' then
		for _, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = val
		end
	else
		local skel, canvas, n_parts = parse_placeholder_string(mix)
		for key, val in pairs(ctx.params) do
			for idx = 2, n_parts, 2 do
				if skel[idx] then canvas[idx] = val
				else canvas[idx] = tostring(key) end
			end
			cache[get_parameter_name(table.concat(canvas))] = val
		end
	end
	ctx.params = cache
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|renaming_to_values|pipe to
--[[
library.renaming_to_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for _, val in pairs(ctx.params) do cache[val] = val end
	ctx.params = cache
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|grouping_by_calling|template
--            name|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.grouping_by_calling = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp, argc, tbl, mem = load_child_opts(ctx.pipe, 2, 0, ctx.params)
	local gargs = {}
	for key, val in pairs(tmp) do
		if type(key) == 'number' and key &lt; 1 then gargs[key - 1] = val
		else gargs[key] = val end
	end
	tmp = ctx.pipe[1]
	if tmp ~= nil then tmp = tmp:match'^%s*(.*%S)' end
	if tmp == nil then error(modulename ..
		', ‘grouping_by_calling’: No template name was provided', 0) end
	local model = { title = tmp }
	local groups = make_groups(tbl)
	for gid, group in pairs(groups) do
		for key, val in pairs(gargs) do group[key] = val end
		group[0], model.args = gid, group
		groups[gid] = ctx.frame:expandTemplate(model)
	end
	for key, val in pairs(mem) do groups[key] = val end
	ctx.params = groups
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|parsing|string to parse|[trim flag]|[iteration
--            delimiter setter]|[...]|[key-value delimiter setter]|[...]|pipe to
library.parsing = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘parsing’: No string to parse was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '|', '=')
	parse_parameter_string(ctx.params, opts[1], isep, iplain, psep, pplain,
		trimnamed, trimunnamed)
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|reinterpreting|parameter to reinterpret|[trim
--            flag]|[iteration delimiter setter]|[...]|[key-value delimiter
--            setter]|[...]|pipe to
library.reinterpreting = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘reinterpreting’: No parameter to reinterpret was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '|', '=')
	local tbl, tmp = ctx.params, get_parameter_name(opts[1])
	local str = tbl[tmp]
	if str ~= nil then
		tbl[tmp] = nil
		parse_parameter_string(tbl, str, isep, iplain, psep, pplain,
			trimnamed, trimunnamed)
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|evaluating|string to parse|[trim flag]|[iteration
--            delimiter setter]|[...]|[key-value delimiter setter]|[...]|pipe to
library.evaluating = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘evaluating’: No string to parse was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '!', ':')
	if opts[1]:match'^%s*(.*%S)' == nil then
		ctx.pipe = copy_or_ref_table(opts, opts ~= ctx.opipe)
		return context_iterate(ctx, argc)
	end
	local new_opts, cache = {}, {}
	local shift = parse_parameter_string(cache, opts[1], isep, iplain,
		psep, pplain, trimnamed, trimunnamed) - argc
	for key, val in pairs(opts) do
		if type(key) ~= 'number' or key &lt; 1 then new_opts[key] = val
		elseif key &gt;= argc then new_opts[key + shift] = val end
	end
	for key, val in pairs(cache) do new_opts[key] = val end
	ctx.pipe = new_opts
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mixing_names_and_values|mixing string|pipe to
library.mixing_names_and_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘mixing_names_and_values’: No mixing string was provided for parameter names', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘mixing_names_and_values’: No mixing string was provided for parameter values', 0) end
	local tmp
	local mix_k = ctx.pipe[1]:match'^%s*(.-)%s*$'
	local cache, mix_v = {}, ctx.pipe[2]
	if mix_k == '$@' and mix_v == '$@' then
		for _, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = val
		end
	elseif mix_k == '$@' and mix_v == '$#' then
		for key, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = tostring(key)
		end
	elseif mix_k == '$#' and mix_v == '$#' then
		for key in pairs(ctx.params) do cache[key] = tostring(key) end
	else
		local skel_k, cnv_k, n_parts_k = parse_placeholder_string(mix_k)
		local skel_v, cnv_v, n_parts_v = parse_placeholder_string(mix_v)
		for key, val in pairs(ctx.params) do
			tmp = tostring(key)
			for idx = 2, n_parts_k, 2 do
				if skel_k[idx] then cnv_k[idx] = val else cnv_k[idx] = tmp end
			end
			for idx = 2, n_parts_v, 2 do
				if skel_v[idx] then cnv_v[idx] = val else cnv_v[idx] = tmp end
			end
			cache[get_parameter_name(table.concat(cnv_k))] =
				table.concat(cnv_v)
		end
	end
	ctx.params = cache
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|swapping_names_and_values|pipe to
--[[
library.swapping_names_and_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do cache[val] = key end
	ctx.params = cache
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|combining|new parameter name|[sort
--            order]|[with/without flushed glue]|setting directives|...|pipe to
library.combining = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	return context_iterate(ctx, combine_parameters(
		ctx,
		function (key, val, kvs) return key .. kvs .. val end,
		'combining'
	) + 1)
end


-- Syntax:  #invoke:params|combining_values|new parameter name|[sort
--            order]|[with/without flushed glue]|setting directives|...|pipe to
library.combining_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	return context_iterate(ctx, combine_parameters(
		ctx,
		function (key, val, kvs) return val end,
		'combining_values'
	) + 1)
end


-- Syntax:  #invoke:params|combining_by_calling|template name|new parameter
--            name|pipe to
library.combining_by_calling = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tname = ctx.pipe[1]
	if tname ~= nil then tname = tname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_calling’: No template name was provided', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘combining_by_calling’: No parameter name was provided', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[2])] = ctx.frame:expandTemplate{
			title = tname,
			args = ctx.params
		}
	}
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|combining_by_invoking|module name|function name|new
--            parameter name|pipe to
library.combining_by_invoking = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local mname = ctx.pipe[1]
	if mname ~= nil then mname = mname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_invoking’: No module name was provided', 0) end
	local fname = ctx.pipe[2]
	if fname ~= nil then fname = fname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_invoking’: No function name was provided', 0) end
	if ctx.pipe[3] == nil then error(modulename ..
		', ‘combining_by_invoking’: No parameter name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = ctx.params }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘mapping_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[3])] =
			tostring(mfunc(ctx.frame:newChild(model)))
	}
	return context_iterate(ctx, 4)
end


-- Syntax:  #invoke:params|combining_by_magic|parser function|new parameter
--            name|pipe to
library.combining_by_magic = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local magic = ctx.pipe[1]
	if magic ~= nil then magic = magic:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_magic’: No parser function was provided', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘combining_by_magic’: No parameter name was provided', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[2])] =
			ctx.frame:callParserFunction(magic, ctx.params)
	}
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|snapshotting|[maximum number]|pipe to
library.snapshotting = function (ctx)
	return context_iterate(ctx, make_child(ctx, ctx.params, 'snapshotting'))
end


-- Syntax:  #invoke:params|remembering|[maximum number]|pipe to
library.remembering = function (ctx)
	return context_iterate(ctx, make_child(ctx, ctx.oparams, 'remembering'))
end


-- Syntax:  #invoke:params|entering_substack|[new]|pipe to
library.entering_substack = function (ctx)
	local tbl, ncurrparent = ctx.params, ctx.n_parents + 1
	if ctx.parents == nil then ctx.parents = { tbl }
	else ctx.parents[ncurrparent] = tbl end
	ctx.n_parents = ncurrparent
	if ctx.pipe[1] ~= nil and ctx.pipe[1]:match'^%s*new%s*$' then
		ctx.params = {}
		return context_iterate(ctx, 2)
	end
	local currsnap = ctx.n_children
	if currsnap &gt; 0 then
		ctx.params, ctx.children[currsnap], ctx.n_children =
			ctx.children[currsnap], nil, currsnap - 1
	else
		local newparams = {}
		for key, val in pairs(tbl) do newparams[key] = val end
		ctx.params = newparams
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|pulling|parameter name|pipe to
library.pulling = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘pulling’: No parameter to pull was provided', 0) end
	local tmp = ctx.n_parents
	local parent = tmp &lt; 1 and ctx.oparams or ctx.parents[tmp]
	tmp = get_parameter_name(opts[1])
	if parent[tmp] ~= nil then ctx.params[tmp] = parent[tmp] end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|recalling|parameter name|pipe to
library.recalling = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘recalling’: No parameter to recall was provided', 0) end
	local arg = get_parameter_name(opts[1])
	if ctx.oparams[arg] ~= nil then ctx.params[arg] = ctx.oparams[arg] end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|finding|parameter name|pipe to
--[[
library.finding = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘finding’: No parameter to find was provided', 0) end
	local arg = get_parameter_name(opts[1])
	local parent
	for idx = ctx.n_parents, 1, -1 do
		parent = ctx.parents[idx]
		if parent[arg] ~= nil then
			ctx.params[arg] = parent[arg]
			return context_iterate(ctx, 2)
		end
	end
	if ctx.oparams[arg] ~= nil then ctx.params[arg] = ctx.oparams[arg] end
	return context_iterate(ctx, 2)
end
]]--


-- Syntax:  #invoke:params|picking|number of parameters to pick|pipe to
--[[
library.picking = function (ctx)
	local len = tonumber(ctx.pipe[1])
	if len == nil or len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘picking’: The number of parameters to pick must be an integer greater than zero', 0) end
	if ctx.n_parents &lt; 1 then copy_table_maxn(ctx.params, ctx.oparams, len)
	else copy_table_maxn(ctx.params, ctx.parents[ctx.n_parents], len) end
	return context_iterate(ctx, 2)
end
]]--


-- Syntax:  #invoke:params|detaching_substack|pipe to
library.detaching_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘detaching_substack’: No substack has been created', 0) end
	local parent = ctx.parents[ncurrparent]
	for key in pairs(ctx.params) do parent[key] = nil end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|dropping_substack|pipe to
library.dropping_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘dropping_substack’: No substack has been created', 0) end
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents =
		ctx.parents[ncurrparent], nil, ncurrparent - 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|leaving_substack|pipe to
library.leaving_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘leaving_substack’: No substack has been created', 0) end
	local currsnap = ctx.n_children + 1
	if ctx.children == nil then ctx.children = { ctx.params }
	else ctx.children[currsnap] = ctx.params end
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents, ctx.n_children =
		ctx.parents[ncurrparent], nil, ncurrparent - 1, currsnap
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|merging_substack|pipe to
library.merging_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘merging_substack’: No substack has been created', 0) end
	local parent, child = ctx.parents[ncurrparent], ctx.params
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents = parent, nil,
		ncurrparent - 1
	for key, val in pairs(child) do parent[key] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|flushing|pipe to
library.flushing = function (ctx)
	if ctx.n_children &lt; 1 then error(modulename ..
		', ‘flushing’: There are no substacks to flush', 0) end
	local parent, currsnap = ctx.params, ctx.n_children
	for key, val in pairs(ctx.children[currsnap]) do parent[key] = val end
	ctx.children[currsnap], ctx.n_children = nil, currsnap - 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|setting_by_flushing|pipe to
library.setting_by_flushing = function (ctx)
	set_strings_from_substack(ctx, ctx, 'setting_by_flushing')
	return context_iterate(ctx, 1)
end


	--[[ Functions ]]--
	-----------------------------


-- Syntax:  #invoke:params|count
library.count = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local retval = 0
	for _ in ctx.iterfunc(ctx.params) do retval = retval + 1 end
	if ctx.subset == -1 then retval = retval - #ctx.params end
	ctx.text = retval
	return false
end


-- Syntax:  #invoke:args|concat_and_call|template name|[prepend 1]|[prepend 2]
--            |[...]|[item n]|[named item 1=value 1]|[...]|[named item n=value
--            n]|[...]
library.concat_and_call = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘concat_and_call’: No template name was provided', 0) end
	remove_numeric_keys(opts, 1, 1)
	ctx.text = ctx.frame:expandTemplate{
		title = tname,
		args = concat_params(ctx)
	}
	return false
end


-- Syntax:  #invoke:args|concat_and_invoke|module name|function name|[prepend
--            1]|[prepend 2]|[...]|[item n]|[named item 1=value 1]|[...]|[named
--            item n=value n]|[...]
library.concat_and_invoke = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘concat_and_invoke’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘concat_and_invoke’: No function name was provided', 0) end
	remove_numeric_keys(opts, 1, 2)
	local mfunc = require('Module:' .. mname)[fname]
	if mfunc == nil then error(modulename ..
		', ‘concat_and_invoke’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	ctx.text = mfunc(ctx.frame:newChild{
		title = 'Module:' .. mname,
		args = concat_params(ctx)
	})
	return false
end


-- Syntax:  #invoke:args|concat_and_magic|parser function|[prepend 1]|[prepend
--            2]|[...]|[item n]|[named item 1=value 1]|[...]|[named item n=
--            value n]|[...]
library.concat_and_magic = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘concat_and_magic’: No parser function was provided', 0) end
	remove_numeric_keys(opts, 1, 1)
	ctx.text = ctx.frame:callParserFunction(magic, concat_params(ctx))
	return false
end


-- Syntax:  #invoke:params|value_of|parameter name
library.value_of = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘value_of’: No parameter name was provided', 0) end
	local val
	local key = opts[1]:match'^%s*(.-)%s*$'
	if key == '0' or key:find'^%-?[1-9]%d*$' ~= nil then
		key = tonumber(key)
		val = ctx.params[key]
		-- No worries: #ctx.params is unused when the modifier is in
		-- first position (and therefore `ctx.params` is a metatable)
		if val ~= nil and (
			ctx.subset ~= -1 or key &gt; #ctx.params or key &lt; 1
		) and (
			ctx.subset ~= 1 or (key &lt;= #ctx.params and key &gt; 0)
		) then
			ctx.text = (ctx.header or '') .. val .. (ctx.footer or '')
		else ctx.text = ctx.ifngiven or '' end
	else
		val = ctx.params[key]
		if ctx.subset ~= 1 and val ~= nil then ctx.text = (ctx.header
			or '') .. val .. (ctx.footer or '')
		else ctx.text = ctx.ifngiven or '' end
	end
	return false
end


-- Syntax:  #invoke:params|list
library.list = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, kvs, pps = {}, 0, ctx.pairsep or '', ctx.itersep or ''
	flush_params(ctx, function (key, val)
		ret[nss + 1], ret[nss + 2], ret[nss + 3], ret[nss + 4], nss =
			pps, key, kvs, val, nss + 4
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 4)
	return false
end


-- Syntax:  #invoke:params|list_values
library.list_values = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	-- NOTE: `library.coins()` and `library.unique_coins()` rely on us
	local ret, nss, pps = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		ret[nss + 1], ret[nss + 2], nss = pps, val, nss + 2
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|list_maybe_with_names
library.list_maybe_with_names = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, kvs, pps = {}, 0, ctx.pairsep or '', ctx.itersep or ''
	mixed_flush_params(
		ctx,
		function (key, val)
			ret[nss + 1], ret[nss + 2], ret[nss + 3],
				ret[nss + 4], nss = pps, '', '', val, nss + 4
		end,
		function (key, val)
			ret[nss + 1], ret[nss + 2], ret[nss + 3],
				ret[nss + 4], nss = pps, key, kvs, val, nss + 4
		end
	)
	finalize_and_return_concatenated_list(ctx, ret, nss, 4)
	return false
end


-- Syntax:  #invoke:params|coins|[first coin = value 1]|[second coin = value
--            2]|[...]|[last coin = value N]
--[[
library.coins = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local opts, tbl = ctx.pipe, ctx.params
	for key, val in pairs(tbl) do tbl[key] = opts[get_parameter_name(val)] end
	return library.list_values(ctx)
end
]]--


-- Syntax:  #invoke:params|unique_coins|[first coin = value 1]|[second coin =
--            value 2]|[...]|[last coin = value N]
--[[
library.unique_coins = function (ctx)
	local tmp
	local opts, tbl = ctx.pipe, ctx.params
	for key, val in pairs(tbl) do
		tmp = get_parameter_name(val)
		tbl[key], opts[tmp] = opts[tmp], nil
	end
	return library.list_values(ctx)
end
]]


-- Syntax:  #invoke:params|for_each|wikitext
library.for_each = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, pps, txt = {}, 0, ctx.itersep or '', ctx.pipe[1] or ''
	local skel, cnv, n_parts = parse_placeholder_string(txt)
	flush_params(ctx, function (key, val)
		for idx = 2, n_parts, 2 do
			if skel[idx] then cnv[idx] = val
			else cnv[idx] = tostring(key) end
		end
		ret[nss + 1], nss = pps, nss + 2
		ret[nss] = table.concat(cnv)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each|template name|[append 1]|[append 2]
--            |[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘call_for_each’: No template name was provided', 0) end
	local model = { title = tname, args = opts }
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	table.insert(opts, 1, true)
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = ctx.frame:expandTemplate(model)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|invoke_for_each|module name|module function|[append
--            1]|[append 2]|[...]|[append n]|[named param 1=value 1]|[...]
--            |[named param n=value n]|[...]
library.invoke_for_each = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘invoke_for_each’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘invoke_for_each’: No function name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = opts }
	local mfunc = require(model.title)[fname]
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = mfunc(ctx.frame:newChild(model))
		
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|magic_for_each|parser function|[append 1]|[append 2]
--            |[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.magic_for_each = function (ctx)
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘magic_for_each’: No parser function was provided', 0) end
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	table.insert(opts, 1, true)
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = ctx.frame:callParserFunction(magic, opts)

	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each_value|template name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each_value = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘call_for_each_value’: No template name was provided', 0) end
	local model = { title = tname, args = opts }
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = ctx.frame:expandTemplate(model)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|invoke_for_each_value|module name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.invoke_for_each_value = function (ctx)
	local opts = ctx.pipe
	local mname, fname
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘invoke_for_each_value’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘invoke_for_each_value’: No function name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = opts }
	local mfunc = require(model.title)[fname]
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	remove_numeric_keys(opts, 1, 1)
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = mfunc(ctx.frame:newChild(model))
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|magic_for_each_value|parser function|[append 1]
--            |[append 2]|[...]|[append n]|[named param 1=value 1]|[...]|[named
--            param n=value n]|[...]
library.magic_for_each_value = function (ctx)
	local opts = ctx.pipe
	local magic
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘magic_for_each_value’: No parser function was provided', 0) end
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = ctx.frame:callParserFunction(magic, opts)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each_group|template name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each_group = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local tmp
	if ctx.pipe[1] ~= nil then tmp = ctx.pipe[1]:match'^%s*(.*%S)' end
	if tmp == nil then error(modulename ..
		', ‘call_for_each_group’: No template name was provided', 0) end
	local model = { title = tmp }
	local opts, ret, nss, ccs = {}, {}, 0, ctx.itersep or ''
	for key, val in pairs(ctx.pipe) do
		if type(key) == 'number' then opts[key - 1] = val
		else opts[key] = val end
	end
	ctx.pipe = opts
	ctx.params = make_groups(ctx.params)
	flush_params(ctx, function (gid, group)
		for key, val in pairs(opts) do group[key] = val end
		group[0], model.args, ret[nss + 1], nss = gid, group, ccs,
			nss + 2
		ret[nss] = ctx.frame:expandTemplate(model)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


	--[[ First-position-only modifiers ]]--
	---------------------------------------


-- Syntax:  #invoke:params|new|pipe to
static_iface.new = function (child_frame)
	local ctx = context_new(child_frame)
	ctx.pipe = copy_or_ref_table(ctx.opipe, false)
	ctx.params = {}
	main_loop(ctx, context_iterate(ctx, 1))
	return ctx.text
end


	--[[ First-position-only functions ]]--
	---------------------------------------


-- Syntax:  #invoke:params|self
static_iface.self = function (frame)
	return frame:getParent():getTitle()
end


	--[[ Public metatable of functions ]]--
	---------------------------------------


return setmetatable({}, {
	__index = function (_, query)
		local fname = query:match'^%s*(.*%S)'
		if fname == nil then error(modulename ..
			': You must specify a function to call', 0) end
		local func = static_iface[fname]
		if func ~= nil then return func end
		func = library[fname]
		if func == nil then error(modulename ..
			': The function ‘' .. fname .. '’ does not exist', 0) end
		return function (child_frame)
			local ctx = context_new(child_frame)
			ctx.pipe = copy_or_ref_table(ctx.opipe, refpipe[fname])
			ctx.params = copy_or_ref_table(ctx.oparams, refparams[fname])
			main_loop(ctx, func)
			return ctx.text
		end
	end
})</text>
      <sha1>s0wwht3ty2l3u443tnfm9mh3z5pmpgj</sha1>
    </revision>
    <revision>
      <id>2819207</id>
      <parentid>2819136</parentid>
      <timestamp>2026-07-24T03:48:20Z</timestamp>
      <contributor>
        <username>Grufo</username>
        <id>1192007</id>
      </contributor>
      <comment>Update from [[d:Special:GoToLinkedPage/mediawikiwiki/Q122696746|master]] using [[mw:Synchronizer| #Synchronizer]]</comment>
      <origin>2819207</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="88044" sha1="9ko1dij40e2ajwgg59fvqm72uoivi2f" xml:space="preserve">require[[strict]]


	---                                        ---
	---     PRIVATE ENVIRONMENT                ---
	---    ________________________________    ---
	---                                        ---


	--[[ Abstract utilities ]]--
	----------------------------


-- Helper function for `string.gsub()` (for managing zero-padded numbers)
local function zero_padded (str)
	return ('%03d%s'):format(#str, str)
end


-- Helper function for `table.sort()` (for natural sorting)
local function natural_sort (var1, var2)
	return var1:gsub('%d+', zero_padded) &lt; var2:gsub('%d+', zero_padded)
end


-- Return a copy or a reference to a table
local function copy_or_ref_table (src, refonly)
	if refonly then return src end
	local newtab = {}
	for key, val in pairs(src) do newtab[key] = val end
	return newtab
end


-- Copy at most N items (of all kinds) from `src` to `dest` and return `dest`
local function copy_table_maxn (dest, src, len)
	local idx = 1
	for key, val in pairs(src) do
		dest[key], idx = val, idx + 1
		if idx &gt; len then break end
	end
	return dest
end


-- Remove some numeric elements from a table, shifting everything to the left
local function remove_numeric_keys (tbl, idx, len)
	local cache, tmp = {}, idx + len - 1
	for key, val in pairs(tbl) do
		if type(key) == 'number' and key &gt;= idx then
			if key &gt; tmp then cache[key - len] = val end
			tbl[key] = nil
		end
	end
	for key, val in pairs(cache) do tbl[key] = val end
end


-- Make a reduced copy of a table (shifting in both directions if necessary)
local function copy_table_reduced (tbl, idx, len)
	local ret, tmp = {}, idx + len - 1
	if idx &gt; 0 then
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &lt; idx then
				ret[key] = val
			elseif key &gt; tmp then ret[key - len] = val end
		end
	elseif tmp &gt; 0 then
		local nshift = 1 - idx
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' then ret[key] = val
			elseif key &gt; tmp then ret[key - tmp] = val
			elseif key &lt; idx then ret[key + nshift] = val end
		end
	else
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &gt; tmp then
				ret[key] = val
			elseif key &lt; idx then ret[key + len] = val end
		end
	end
	return ret
end


-- Make an expanded copy of a table (shifting in both directions if necessary)
local function copy_table_expanded (tbl, idx, len)
	local ret, tmp = {}, idx + len - 1
	if idx &gt; 0 then
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &lt; idx then
				ret[key] = val
			else ret[key + len] = val end
		end
	elseif tmp &gt; 0 then
		local nshift = idx - 1
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' then ret[key] = val
			elseif key &gt; 0 then ret[key + tmp] = val
			elseif key &lt; 1 then ret[key + nshift] = val end
		end
	else
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &gt; tmp then
				ret[key] = val
			else ret[key - len] = val end
		end
	end
	return ret
end


-- Given a table, create two new tables containing the sorted list of keys
local function get_key_list_sorted (tbl, sort_fn)
	local nums, words, nn, nw = {}, {}, 0, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			nn = nn + 1
			nums[nn] = key
		else
			nw = nw + 1
			words[nw] = key
		end
	end
	table.sort(nums)
	table.sort(words, sort_fn)
	return nums, words, nn, nw
end


-- Parse a parameter name string and return it as a string or a number
local function get_parameter_name (par_str)
	local ret = par_str:match'^%s*(.-)%s*$'
	if ret ~= '0' and ret:find'^%-?[1-9]%d*$' == nil then return ret end
	return tonumber(ret)
end


-- Move a key from a table to another, but only if under a different name and
-- always parsing numeric strings as numbers
local function steal_if_renamed (val, src, skey, dest, dkey)
	local realkey = get_parameter_name(dkey)
	if skey ~= realkey then dest[realkey], src[skey] = val, nil end
end


	--[[ Public strings ]]--
	------------------------


-- Special match keywords (functions and modifiers MUST avoid these names)
local mkeywords = {
	['or'] = 0,
	pattern = 1,
	plain = 2,
	strict = 3
}


-- Sort functions (functions and modifiers MUST avoid these names)
local sortfunctions = {
	alphabetically = false,
	naturally = natural_sort
}


-- Callback styles for the `mapping_*` and `renaming_*` class of modifiers
-- (functions and modifiers MUST avoid these names)
--[[

Meanings of the columns:

  col[1] = Loop type (0-3)
  col[2] = Number of module arguments that the style requires (1-3)
  col[3] = Minimum number of sequential parameters passed to the callback
  col[4] = Name of the callback parameter where to place each parameter name
  col[5] = Name of the callback parameter where to place each parameter value
  col[6] = Argument in the modifier's invocation that will override `col[4]`
  col[7] = Argument in the modifier's invocation that will override `col[5]`

A value of `-1` indicates that no meaningful value is stored (i.e. `nil`)

]]--
local mapping_styles = {
	names_and_values = { 3, 2, 2, 1, 2, -1, -1 },
	values_and_names = { 3, 2, 2, 2, 1, -1, -1 },
	values_only = { 1, 2, 1, -1, 1, -1, -1 },
	names_only = { 2, 2, 1, 1, -1, -1, -1 },
	names_and_values_as = { 3, 4, 0, -1, -1, 2, 3 },
	names_only_as = { 2, 3, 0, -1, -1, 2, -1 },
	values_only_as = { 1, 3, 0, -1, -1, -1, 2 },
	blindly = { 0, 2, 0, -1, -1, -1, -1 }
}


-- Memory slots (functions and modifiers MUST avoid these names)
local memoryslots = {
	h = 'header',
	f = 'footer',
	i = 'itersep',
	l = 'lastsep',
	n = 'ifngiven',
	p = 'pairsep',
	s = 'oxfordsep'
}


-- Possible trimming modes for the `parsing` modifier
local trim_parse_opts = {
	trim_none = { false, false },
	trim_positional = { false, true },
	trim_named = { true, false },
	trim_all = { true, true }
}


-- Possible string modes for the iteration separator in the `parsing` and
-- `reinterpreting` modifiers
local isep_parse_opts = {
	splitter_pattern = false,
	splitter_string = true
}


-- Possible string modes for the key-value separator in the `parsing` and
-- `reinterpreting` modifiers
local psep_parse_opts = {
	setter_pattern = false,
	setter_string = true
}


-- Possible position references for the `splicing` modifier
local position_references = {
	add_nothing = 0,
	add_smallest_number = 1,
	add_last_of_sequence = 2,
	add_largest_number = 3
}


-- Possible modes for the `reassigning` modifier
local a_modes = {
	transfer = 0,
	clone = 1,
	rename = 2,
	copy = 3,
	sacrifice = 4,
	provide = 5,
	spare = 7
}


-- Functions and modifiers MUST avoid these names too: `here`, `in_substack`,
-- `let`, `expose`, `use`, `with_flushed_glue`, `without_flushed_glue`
-- `without_sorting`


	--[[ Private constants ]]--
	---------------------------


-- Hard-coded name of the module (to avoid going through `frame:getTitle()`)
local modulename = 'Module:Params'


-- The functions listed here declare that they don't need the `frame.args`
-- metatable to be copied into a regular table; if they are modifiers they also
-- guarantee that they will make their own (modified) copy available
local refpipe = {
	call_for_each_group = true,
	--coins = true,
	count = true,
	evaluating = true,
	for_each = true,
	list = true,
	list_values = true,
	list_maybe_with_names = true,
	value_of = true
}


-- The functions listed here declare that they don't need the
-- `frame:getParent().args` metatable to be copied into a regular table; if 
-- they are modifiers they also guarantee that they will make their own
-- (modified) copy available
local refparams = {
	call_for_each_group = true,
	combining = true,
	combining_by_calling = true,
	combining_values = true,
	concat_and_call = true,
	concat_and_invoke = true,
	concat_and_magic = true,
	count = true,
	grouping_by_calling = true,
	mixing_names_and_values = true,
	keeping_at_most = true,
	renaming_by_mixing = true,
	renaming_to_sequence = true,
	renaming_to_uppercase = true,
	renaming_to_lowercase = true,
	--renaming_to_values = true,
	shifting = true,
	splicing = true,
	--swapping_names_and_values = true,
	value_of = true,
	with_name_matching = true
}


-- Maximum number of numeric parameters that can be filled, if missing (we
-- chose an arbitrary number for this constant; you can discuss about its
-- optimal value at Module talk:Params)
local maxfill = 1024


-- The private table of functions
local library = {}


-- Functions and modifiers that can only be invoked in first position
local static_iface = {}


	--[[ Private functions ]]--
	---------------------------


-- Create a new context
local function context_new (child_frame)
	local main_frame = child_frame:getParent()
	return {
		frame = main_frame,
		opipe = child_frame.args,
		oparams = main_frame.args,
		firstposonly = static_iface,
		iterfunc = pairs,
		sorttype = 0,
		n_parents = 0,
		n_children = 0,
		n_available = maxfill
	}
end


-- Move to the next action within the user-given list
local function context_iterate (ctx, n_forward)
	local nextfn
	if ctx.pipe[n_forward] ~= nil then
		nextfn = ctx.pipe[n_forward]:match'^%s*(.*%S)'
	end
	if nextfn == nil then error(modulename ..
		': You must specify a function to call', 0) end
	if library[nextfn] == nil then
		if ctx.firstposonly[nextfn] == nil then error(modulename ..
			': The function ‘' .. nextfn .. '’ does not exist', 0)
		else error(modulename .. ': The ‘' .. nextfn ..
			'’ directive can only appear in first position', 0)
		end
	end
	remove_numeric_keys(ctx.pipe, 1, n_forward)
	return library[nextfn]
end


-- Main loop
local function main_loop (ctx, start_with)
	local fn = start_with
	repeat fn = fn(ctx) until not fn
	if ctx.n_parents &gt; 0 then error(modulename ..
		': One or more ‘merging_substack’ directives are missing', 0) end
	if ctx.n_children &gt; 0 then error(modulename ..
		', For some of the snapshots either the ‘flushing’ directive is missing or a group has not been properly closed with ‘merging_substack’', 0) end
end


-- Load a `setting`-like directive string into the `dest` table
local function set_strings_from_opts (dest, opts, start_from)
	local cmd
	if opts[start_from] == nil then return start_from - 1 end
	cmd = opts[start_from]:gsub('%s+', ''):gsub('/+', '/')
		:match'^/*(.*[^/])'
	if cmd == nil then return start_from end
	local vname, chr
	local amap, sep, argc = {}, string.byte('/'), start_from + 1
	for idx = 1, #cmd do
		chr = cmd:byte(idx)
		if chr == sep then
			for key, val in ipairs(amap) do
				dest[val], amap[key] = opts[argc], nil
			end
			argc = argc + 1
		else
			vname = memoryslots[string.char(chr)]
			if vname == nil then error(modulename ..
				', ‘setting’: Unknown slot ‘' ..
				string.char(chr) .. '’', 0) end
			table.insert(amap, vname)
		end
	end
	for key, val in ipairs(amap) do dest[val] = opts[argc] end
	return argc
end


-- Add a new stack of parameters to `ctx.children`
local function new_substack (ctx)
	local currsnap, newparams = ctx.n_children + 1, {}
	if ctx.children == nil then ctx.children = { newparams }
	else ctx.children[currsnap] = newparams end
	ctx.n_children = currsnap
	return newparams
end


-- Parse a raw argument containing a `sortfunctions` directive, or
-- `'without_sorting'`, or `nil`
local function load_sort_opt (raw_arg)
	if raw_arg == nil then return nil, 1, false end
	local trarg = raw_arg:match'^%s*(.-)%s*$'
	if trarg == 'without_sorting' then return nil, 2, false, trarg end
	local tmp = sortfunctions[trarg]
	if tmp == nil then return nil, 1, false, trarg end
	return tmp or nil, 2, true, trarg
end


-- Parse optional user arguments of type `...|[let/use]|[...]|[let/use]|[...]|
-- [number of additional parameters]|[parameter 1]|[parameter 2]|[...]`
local function load_child_opts (src, start_from, append_after, params)
	local tnamed, tmp1, tmp2
	local pin, tbl, mem = start_from, {}, {}
	while src[pin] ~= nil and src[pin + 1] ~= nil do
		tmp1 = src[pin]:match'^%s*(.*%S)'
		if tmp1 == 'let' and src[pin + 2] ~= nil then
			tmp1 = get_parameter_name(src[pin + 1])
			mem[tmp1], tbl[tmp1], pin = nil, src[pin + 2], pin + 3
--[[
		elseif tmp1 == 'expose' then
			tmp1 = get_parameter_name(src[pin + 1])
			tmp2 = params[tmp1]
			mem[tmp1], tbl[tmp1], pin = tmp2, tmp2, pin + 2
]]--
		elseif tmp1 == 'use' and src[pin + 2] ~= nil then
			tmp1 = get_parameter_name(src[pin + 2])
			tmp2 = params[tmp1]
			mem[tmp1], tbl[get_parameter_name(src[pin + 1])], pin =
				tmp2, tmp2, pin + 3
		else break end
	end
	local tnew = copy_or_ref_table(params, next(mem) == nil)
	for key in pairs(mem) do tnew[key] = nil end
	if pin ~= start_from then tnamed, tbl = tbl, {} end
	tmp1 = tonumber(src[pin])
	if tmp1 ~= nil and math.floor(tmp1) == tmp1 then
		if tmp1 &lt; 0 then tmp1 = -1 end
		tmp2 = append_after - pin
		for idx = pin + 1, pin + tmp1 do tbl[idx + tmp2] = src[idx] end
		pin = pin + tmp1 + 1
	end
	if tnamed ~= nil then
		for key, val in pairs(tnamed) do tbl[key] = val end
	end
	return tbl, pin, tnew, mem
end


-- Load the optional arguments of some of the `mapping_*` and `renaming_*`
-- class of modifiers
local function load_callback_opts (src, n_skip, default_style, params)
	local style, shf
	local tmp = src[n_skip + 1]
	if tmp ~= nil then style = mapping_styles[tmp:match'^%s*(.-)%s*$'] end
	if style == nil then style, shf = default_style, n_skip - 1
	else shf = n_skip end
	local n_exist, karg, varg = style[3], style[4], style[5]
	tmp = style[6]
	if tmp &gt; -1 then
		karg = src[tmp + shf]:match'^%s*(.-)%s*$'
		if karg == '0' or karg:find'^%-?[1-9]%d*$' ~= nil then
			karg = tonumber(karg)
			n_exist = math.max(n_exist, karg)
		end
	end
	tmp = style[7]
	if tmp &gt; -1 then
		varg = src[tmp + shf]:match'^%s*(.-)%s*$'
		if varg == '0' or varg:find'^%-?[1-9]%d*$' ~= nil then
			varg = tonumber(varg)
			n_exist = math.max(n_exist, varg)
		end
	end
	local dest, argc, tnew, mem = load_child_opts(src, style[2] + shf,
		n_exist, params)
	tmp = style[1]
	if (tmp == 3 or tmp == 2) and dest[karg] ~= nil then
		tmp = tmp - 2 end
	if (tmp == 3 or tmp == 1) and dest[varg] ~= nil then
		tmp = tmp - 1 end
	return dest, argc, tmp, karg, varg, tnew, mem
end


-- Parse the arguments of some of the `mapping_*` and `renaming_*` class of
-- modifiers
local function load_replace_args (opts, whoami)
	if opts[1] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No pattern string was given', 0) end
	if opts[2] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No replacement string was given', 0) end
	local ptn, repl, nmax, argc = opts[1], opts[2], tonumber(opts[3]), 3
	if nmax ~= nil or (opts[3] or ''):match'^%s*$' ~= nil then argc = 4 end
	local flg = opts[argc]
	if flg ~= nil then flg = mkeywords[flg:match'^%s*(.-)%s*$'] end
	if flg == 0 then flg = nil elseif flg ~= nil then argc = argc + 1 end
	return ptn, repl, nmax, flg, argc, (nmax ~= nil and nmax &lt; 1) or
		(flg == 3 and ptn == repl)
end


-- Parse the arguments of the `with_*_matching` class of modifiers
local function load_pattern_args (opts, whoami)
	local keyw
	local ptns, state, nptns, cnt = {}, 0, 0, 1
	for _, val in ipairs(opts) do
		if state == 0 then
			nptns, state = nptns + 1, -1
			ptns[nptns] = { val, false, false }
		else
			keyw = val:match'^%s*(.*%S)'
			if keyw == nil or mkeywords[keyw] == nil or (
				state &gt; 0 and mkeywords[keyw] &gt; 0
			) then break
			else
				state = mkeywords[keyw]
				if state &gt; 1 then ptns[nptns][2] = true end
				if state == 3 then ptns[nptns][3] = true end
			end
		end
		cnt = cnt + 1
	end
	if state == 0 then error(modulename .. ', ‘' .. whoami ..
		'’: No pattern was given', 0) end
	return ptns, nptns, cnt
end


-- Load the optional arguments of the `parsing`, `reinterpreting` and
-- `evaluating` modifiers
local function load_parse_opts (opts, start_from, isp, psp)
	local tmp
	local optslots, noptslots, argc, trimn, trimu, iplain, pplain =
		{ true, true, true }, 3, start_from, true, false, true, true
	repeat
		noptslots, tmp = noptslots - 1, opts[argc]
		if tmp == nil then break end
		tmp = tmp:match'^%s*(.-)%s*$'
		if optslots[1] ~= nil and trim_parse_opts[tmp] ~= nil then
			tmp = trim_parse_opts[tmp]
			trimn, trimu, optslots[1] = tmp[1], tmp[2], nil
		elseif optslots[2] ~= nil and isep_parse_opts[tmp] ~= nil then
			argc = argc + 1
			iplain, isp, optslots[2] = isep_parse_opts[tmp],
				opts[argc], nil
		elseif optslots[3] ~= nil and psep_parse_opts[tmp] ~= nil then
			argc = argc + 1
			pplain, psp, optslots[3] = psep_parse_opts[tmp],
				opts[argc], nil
		else break end
		argc = argc + 1
	until noptslots &lt; 1
	return isp, iplain, psp, pplain, trimn, trimu, argc
end


-- Map parameters' values using a custom callback and a referenced table
local value_maps = {
	[0] = function (tbl, margs, karg, varg, fn)
		for key in pairs(tbl) do tbl[key] = fn() end
	end,
	[1] = function (tbl, margs, karg, varg, fn)
		for key, val in pairs(tbl) do
			margs[varg] = val
			tbl[key] = fn()
		end
	end,
	[2] = function (tbl, margs, karg, varg, fn)
		for key in pairs(tbl) do
			margs[karg] = key
			tbl[key] = fn()
		end
	end,
	[3] = function (tbl, margs, karg, varg, fn)
		for key, val in pairs(tbl) do
			margs[karg], margs[varg] = key, val
			tbl[key] = fn()
		end
	end
}


-- Private table for `map_names()`
local name_thieves = {
	[0] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[1] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[varg] = val
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[2] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[karg] = key
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[3] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[karg], rargs[varg] = key, val
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end
}


-- Map parameters' names using a custom callback and a referenced table
local function map_names (tbl, rargs, karg, varg, looptype, fn)
	local cache = {}
	name_thieves[looptype](cache, tbl, rargs, karg, varg, fn)
	for key, val in pairs(cache) do tbl[key] = val end
end


-- Return a new table that contains `src` regrouped according to the numeric
-- suffixes in its keys
local function make_groups (src)
	-- NOTE: `src` might be the original metatable!
	local prefix, gid
	local groups = {}
	for key, val in pairs(src) do
		-- `key` must only be a string or a number...
		if type(key) == 'string' then
			prefix, gid = key:match'^%s*(.-)%s*(%-?%d*)%s*$'
			gid = tonumber(gid) or ''
		else
			prefix, gid = '', key
		end
		if groups[gid] == nil then groups[gid] = {} end
		if prefix == '0' or prefix:find'^%-?[1-9]%d*$' ~= nil then
			prefix = tonumber(prefix)
			if prefix &lt; 1 then prefix = prefix - 1 end
		end
		groups[gid][prefix] = val
	end
	return groups
end


-- Split into parts a string containing the `$#` and `$@` placeholders and
-- return the information as a skeleton table, a canvas table and a length
local function parse_placeholder_string (target)
	local idx, s_pos, skel, canvas = 1, 1, {}, {}
	local e_pos = string.find(target, '%$[@#]', 1, false)
	while e_pos ~= nil do
		canvas[idx] = target:sub(s_pos, e_pos - 1)
		skel[idx + 1] = target:sub(e_pos, e_pos + 1) == '$@'
		idx = idx + 2
		s_pos = e_pos + 2
		e_pos = string.find(target, '%$[@#]', s_pos, false)
	end
	if (s_pos &gt; target:len()) then idx = idx - 1
	else canvas[idx] = target:sub(s_pos) end
	return skel, canvas, idx
end


-- Populate a table by parsing a parameter string (heavy lifting for `parsing`,
-- `reinterpreting` and `evaluating`)
local function parse_parameter_string (tbl, str, isp, ipl, psp, ppl, trn, tru)
	local key, val, spos1, spos2, pos1, pos2
	local pos3, idx, lenplone = 0, 1, #str + 1
	if isp == nil or isp == '' then
		if psp == nil or psp == '' then
			if tru then tbl[idx] = str:match'^%s*(.-)%s*$'
			else tbl[idx] = str end
			return idx
		end
		spos1, spos2 = str:find(psp, 1, ppl)
		if spos1 == nil then
			key = idx
			if tru then val = str:match'^%s*(.-)%s*$'
			else val = str end
			idx = idx + 1
		else
			key = get_parameter_name(str:sub(1, spos1 - 1))
			val = str:sub(spos2 + 1)
			if trn then val = val:match'^%s*(.-)%s*$' end
		end
		tbl[key] = val
		return idx
	end
	if psp == nil or psp == '' then
		repeat
			pos1 = pos3 + 1
			pos2, pos3 = str:find(isp, pos1, ipl)
			val = str:sub(pos1, (pos2 or lenplone) - 1)
			if tru then val = val:match'^%s*(.-)%s*$' end
			tbl[idx], idx = val, idx + 1
		until pos2 == nil
		return idx
	end
	repeat
		pos1 = pos3 + 1
		pos2, pos3 = str:find(isp, pos1, ipl)
		val = str:sub(pos1, (pos2 or lenplone) - 1)
		spos1, spos2 = val:find(psp, 1, ppl)
		if spos1 == nil then
			key = idx
			if tru then val = val:match'^%s*(.-)%s*$' end
			idx = idx + 1
		else
			key = get_parameter_name(val:sub(1, spos1 - 1))
			val = val:sub(spos2 + 1)
			if trn then val = val:match'^%s*(.-)%s*$' end
		end
		tbl[key] = val
	until pos2 == nil
	return idx
end


-- Heavy lifting for `snapshotting` and `remembering`
local function make_child (ctx, src, whoami)
	local len = tonumber(ctx.pipe[1])
	if len == nil or len == 0 then
		local stack = new_substack(ctx)
		for key, val in pairs(src) do stack[key] = val end
		return len == nil and 1 or 2
	end
	if len &lt; 0 or math.floor(len) ~= len then error(modulename ..
		', ‘' .. whoami .. '’: The number of parameters to copy must be an integer greater than zero', 0) end
	copy_table_maxn(new_substack(ctx), src, len)
	return 2
end


-- Heavy lifting for `setting_by_flushing`, `combining` and `combining_values`
local function set_strings_from_substack (ctx, dest, whoami)
	if ctx.n_children &lt; 1 then error(modulename ..
		', ‘' .. whoami .. '’: There are no substacks to flush', 0) end
	local currsnap = ctx.n_children
	local stack = ctx.children[currsnap]
	for key, val in pairs(memoryslots) do
		if stack[key] ~= nil then dest[val] = stack[key] end
	end
	ctx.children[currsnap], ctx.n_children = nil, currsnap - 1
end


-- Heavy lifting for `combining` and `combining_values`
local function combine_parameters (ctx, keyval_fn, whoami)
	-- NOTE: `ctx.params` might be the original metatable! This function
	-- MUST create a copy of it before returning
	local opts = ctx.pipe
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No parameter name was provided', 0) end
	local argc
	local tbl, vars = ctx.params, {}
	local sortfn, varsarg0, do_sort, tmp = load_sort_opt(opts[2])
	if varsarg0 == 2 then tmp = opts[3] and opts[3]:match'^%s*(.-)%s*$' end
	if tmp == 'with_flushed_glue' then
		varsarg0 = varsarg0 + 1
		argc = set_strings_from_opts(vars, opts, varsarg0 + 1)
		set_strings_from_substack(ctx, vars, whoami)
	else
		if tmp == 'without_flushed_glue' then varsarg0 = varsarg0 + 1 end
		argc = set_strings_from_opts(vars, opts, varsarg0 + 1)
	end
	if argc &lt; varsarg0 then error(modulename ..
		', ‘' .. whoami .. '’: No setting directive was given', 0) end
	if next(tbl) == nil then
		if vars.ifngiven ~= nil then ctx.params =
			{ [get_parameter_name(ctx.pipe[1])] = vars.ifngiven }
		elseif tbl == ctx.oparams then ctx.params = {} end
		return argc
	end
	local cache, len
	if do_sort then
		local words
		cache, words, len, tmp = get_key_list_sorted(tbl, sortfn)
		for idx = 1, tmp do cache[len + idx] = words[idx] end
		len = len + tmp
	else
		len, cache = 0, {}
		for key in pairs(tbl) do
			len = len + 1
			cache[len] = key
		end
	end
	local pmap, nss, kvs, pps = {}, 0, vars.pairsep or '', vars.itersep or ''
	for idx = 1, len do
		tmp, pmap[nss + 1] = cache[idx], pps
		pmap[nss + 2] = keyval_fn(tmp, tbl[tmp], kvs)
		nss = nss + 2
	end
	tmp = vars.oxfordsep or vars.lastsep
	if tmp ~= nil and nss &gt; 4 then pmap[nss - 1] = tmp
	elseif nss &gt; 2 and vars.lastsep ~= nil then
		pmap[nss - 1] = vars.lastsep
	end
	pmap[1] = vars.header or ''
	if vars.footer ~= nil then pmap[nss + 1] = vars.footer end
	ctx.params = { [get_parameter_name(ctx.pipe[1])] = table.concat(pmap) }
	return argc
end


-- Concatenate the numeric keys from the table of parameters to the numeric
-- keys from the table of options; non-numeric keys from the table of options
-- will prevail over colliding non-numeric keys from the table of parameters
local function concat_params (ctx)
	local retval, tbl, nmax = {}, ctx.params, table.maxn(ctx.pipe)
	if ctx.subset == 1 then
		-- We need only the sequence
		for key, val in ipairs(tbl) do retval[key + nmax] = val end
	else
		if ctx.subset == -1 then
			for key in ipairs(tbl) do tbl[key] = nil end
		end
		for key, val in pairs(tbl) do
			if type(key) == 'number' and key &gt; 0 then
				retval[key + nmax] = val
			else retval[key] = val end
		end
	end
	for key, val in pairs(ctx.pipe) do retval[key] = val end
	return retval
end


-- Flush the parameters by calling a custom function for each value (after this
-- function has been invoked `ctx.params` will be no longer usable)
local function flush_params (ctx, fn)
	local tbl = ctx.params
	if ctx.subset == 1 then
		for key, val in ipairs(tbl) do fn(key, val) end
		return
	end
	if ctx.subset == -1 then
		for key, val in ipairs(tbl) do tbl[key] = nil end
	end
	if ctx.sorttype &gt; 0 then
		local nums, words, nn, nw = get_key_list_sorted(tbl, natural_sort)
		if ctx.sorttype == 2 then
			for idx = 1, nw do fn(words[idx], tbl[words[idx]]) end
			for idx = 1, nn do fn(nums[idx], tbl[nums[idx]]) end
			return
		end
		for idx = 1, nn do fn(nums[idx], tbl[nums[idx]]) end
		for idx = 1, nw do fn(words[idx], tbl[words[idx]]) end
		return
	end
	if ctx.subset ~= -1 then
		for key, val in ipairs(tbl) do
			fn(key, val)
			tbl[key] = nil
		end
	end
	for key, val in pairs(tbl) do fn(key, val) end
end


-- Flush the parameters by calling one of two custom functions for each value
-- (after this function has been invoked `ctx.params` will be no longer usable)
local function mixed_flush_params (ctx, fn_seq, fn_oth)
	if ctx.subset == 1 then
		for key, val in ipairs(ctx.params) do fn_seq(key, val) end
		return
	end
	if ctx.subset == -1 then
		flush_params(ctx, fn_oth)
		return
	end
	local tbl = ctx.params
	if ctx.sorttype &gt; 0 then
		local nums, words, nn, nw = get_key_list_sorted(tbl, natural_sort)
		local sequence = {}
		for key, val in ipairs(tbl) do sequence[key] = val end
		if ctx.sorttype == 2 then
			for idx = 1, nw do fn_oth(words[idx], tbl[words[idx]]) end
		end
		for idx = 1, nn do
			if sequence[nums[idx]] then fn_seq(nums[idx], sequence[nums[idx]])
			else fn_oth(nums[idx], tbl[nums[idx]]) end
		end
		if ctx.sorttype ~= 2 then
			for idx = 1, nw do fn_oth(words[idx], tbl[words[idx]]) end
		end
		return
	end
	for key, val in ipairs(tbl) do
		fn_seq(key, val)
		tbl[key] = nil
	end
	for key, val in pairs(tbl) do fn_oth(key, val) end
end


-- Finalize and return a concatenated list
local function finalize_and_return_concatenated_list (ctx, lst, len, modsize)
	if len &gt; 0 then
		local tmp = ctx.oxfordsep or ctx.lastsep
		if tmp ~= nil and len &gt; modsize * 2 then
			lst[len - modsize + 1] = tmp
		elseif len &gt; modsize and ctx.lastsep ~= nil then
			lst[len - modsize + 1] = ctx.lastsep
		end
		lst[1] = ctx.header or ''
		if ctx.footer ~= nil then lst[len + 1] = ctx.footer end
		ctx.text = table.concat(lst)
	else ctx.text = ctx.ifngiven or '' end
end


	---                                        ---
	---     PUBLIC ENVIRONMENT                 ---
	---    ________________________________    ---
	---                                        ---


	--[[ Modifiers ]]--
	-------------------


-- Syntax:  #invoke:params|sequential|pipe to
library.sequential = function (ctx)
	if ctx.subset == 1 then error(modulename ..
		': The ‘sequential’ directive has been provided more than once', 0) end
	if ctx.subset == -1 then error(modulename ..
		': The two directives ‘non-sequential’ and ‘sequential’ are in contradiction with each other', 0) end
	if ctx.sorttype &gt; 0 then error(modulename ..
		': The ‘all_sorted’ and ‘reassorted’ directives are redundant when followed by ‘sequential’', 0) end
	ctx.iterfunc, ctx.subset = ipairs, 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|non-sequential|pipe to
library['non-sequential'] = function (ctx)
	if ctx.subset == -1 then error(modulename ..
		': The ‘non-sequential’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The two directives ‘sequential’ and ‘non-sequential’ are in contradiction with each other', 0) end
	ctx.iterfunc, ctx.subset = pairs, -1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|all_sorted|pipe to
library.all_sorted = function (ctx)
	if ctx.sorttype == 1 then error(modulename ..
		': The ‘all_sorted’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The ‘all_sorted’ directive is redundant after ‘sequential’', 0) end
	if ctx.sorttype == 2 then error(modulename ..
		': The two directives ‘reassorted’ and ‘sequential’ are in contradiction with each other', 0) end
	ctx.sorttype = 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|reassorted|pipe to
library.reassorted = function (ctx)
	if ctx.sorttype == 2 then error(modulename ..
		': The ‘reassorted’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The ‘reassorted’ directive is redundant after ‘sequential’', 0) end
	if ctx.sorttype == 1 then error(modulename ..
		': The two directives ‘sequential’ and ‘reassorted’ are in contradiction with each other', 0) end
	ctx.sorttype = 2
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|setting|directives|...|pipe to
library.setting = function (ctx)
	local argc = set_strings_from_opts(ctx, ctx.pipe, 1)
	if argc &lt; 2 then error(modulename ..
		', ‘setting’: No directive was given', 0) end
	return context_iterate(ctx, argc + 1)
end


-- Syntax:  #invoke:params|scoring|new parameter name|[container]|pipe to
library.scoring = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘scoring’: No parameter name was provided', 0) end
	local tmp
	local retval, opts = 0, ctx.pipe
	for _ in pairs(ctx.params) do retval = retval + 1 end
	if opts[2] ~= nil then tmp = opts[2]:match'^%s*(.*%S)' end
	if tmp == 'in_substack' then
		new_substack(ctx)[get_parameter_name(opts[1])] = tostring(retval)
		return context_iterate(ctx, 3)
	end
	ctx.params[get_parameter_name(opts[1])] = tostring(retval)
	return context_iterate(ctx, tmp == 'here' and 3 or 2)
end


-- Syntax:  #invoke:params|squeezing|pipe to
library.squeezing = function (ctx)
	local store, indices, tbl, newlen = {}, {}, ctx.params, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			newlen = newlen + 1
			indices[newlen], store[key], tbl[key] = key, val, nil
		end
	end
	table.sort(indices)
	for idx = 1, newlen do tbl[idx] = store[indices[idx]] end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|filling_the_gaps|pipe to
library.filling_the_gaps = function (ctx)
	local newval, tbl, tmp, nmin, nmax, nnums =
		ctx.pipe[1], ctx.params, {}, 1, nil, -1
	if newval == nil then error(modulename ..
		', ‘filling_the_gaps’: No value was provided', 0) end
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			if nmax == nil then
				if key &lt; nmin then nmin = key end
				nmax = key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
			tmp[key], nnums = val, nnums + 1
		end
	end
	if nmax ~= nil and nmax - nmin &gt; nnums then
		ctx.n_available = ctx.n_available + nmin + nnums - nmax
		if ctx.n_available &lt; 0 then error(modulename ..
			', ‘filling_the_gaps’: It is possible to fill at most ' ..
			tostring(maxfill) .. ' parameters', 0) end
		for idx = nmin, nmax, 1 do tbl[idx] = newval end
		for key, val in pairs(tmp) do tbl[key] = val end
	end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|clearing|pipe to
library.clearing = function (ctx)
	local tbl, numerics = ctx.params, {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key], tbl[key] = val, nil
		end
	end
	for key, val in ipairs(numerics) do tbl[key] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|cutting|left cut|right cut|pipe to
library.cutting = function (ctx)
	local lcut = tonumber(ctx.pipe[1])
	if lcut == nil or math.floor(lcut) ~= lcut then error(modulename ..
		', ‘cutting’: Left cut must be an integer number', 0) end
	local rcut = tonumber(ctx.pipe[2])
	if rcut == nil or math.floor(rcut) ~= rcut then error(modulename ..
		', ‘cutting’: Right cut must be an integer number', 0) end
	local tbl = ctx.params
	local len = #tbl
	if lcut &lt; 0 then lcut = len + lcut end
	if rcut &lt; 0 then rcut = len + rcut end
	local tot = lcut + rcut
	if tot &gt; 0 then
		local cache = {}
		if tot &gt;= len then
			for key in ipairs(tbl) do tbl[key] = nil end
			tot = len
		else
			for idx = len - rcut + 1, len, 1 do tbl[idx] = nil end
			for idx = 1, lcut, 1 do tbl[idx] = nil end
		end
		for key, val in pairs(tbl) do
			if type(key) == 'number' and key &gt; 0 then
				if key &gt; len then cache[key - tot] = val
				else cache[key - lcut] = val end
				tbl[key] = nil
			end
		end
		for key, val in pairs(cache) do tbl[key] = val end
	end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|cropping|left crop|right crop|pipe to
library.cropping = function (ctx)
	local lcut = tonumber(ctx.pipe[1])
	if lcut == nil or math.floor(lcut) ~= lcut then error(modulename ..
		', ‘cropping’: Left crop must be an integer number', 0) end
	local rcut = tonumber(ctx.pipe[2])
	if rcut == nil or math.floor(rcut) ~= rcut then error(modulename ..
		', ‘cropping’: Right crop must be an integer number', 0) end
	local tbl = ctx.params
	local nmin, nmax
	for key in pairs(tbl) do
		if type(key) == 'number' then
			if nmin == nil then nmin, nmax = key, key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
		end
	end
	if nmin ~= nil then
		local len = nmax - nmin + 1
		if lcut &lt; 0 then lcut = len + lcut end
		if rcut &lt; 0 then rcut = len + rcut end
		if lcut + rcut - len &gt; -1 then
			for key in pairs(tbl) do
				if type(key) == 'number' then tbl[key] = nil end
			end
		elseif lcut + rcut &gt; 0 then
			for idx = nmax - rcut + 1, nmax do tbl[idx] = nil end
			for idx = nmin, nmin + lcut - 1 do tbl[idx] = nil end
			local lshift = nmin + lcut - 1
			if lshift &gt; 0 then
				for idx = lshift + 1, nmax, 1 do
					tbl[idx - lshift], tbl[idx] =
						tbl[idx], nil
				end
			end
		end
	end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|purging|start offset|length|pipe to
library.purging = function (ctx)
	local idx = tonumber(ctx.pipe[1])
	if idx == nil or math.floor(idx) ~= idx then error(modulename ..
		', ‘purging’: Start offset must be an integer number', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘purging’: Length must be an integer number', 0) end
	local tbl = ctx.params
	if len &lt; 1 then
		len = len + table.maxn(tbl)
		if idx &gt; len then return context_iterate(ctx, 3) end
		len = len - idx + 1
	end
	ctx.params = copy_table_reduced(tbl, idx, len)
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|backpurging|start offset|length|pipe to
library.backpurging = function (ctx)
	local last = tonumber(ctx.pipe[1])
	if last == nil or math.floor(last) ~= last then error(modulename ..
		', ‘backpurging’: Start offset must be an integer number', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘backpurging’: Length must be an integer number', 0) end
	local idx
	local tbl = ctx.params
	if len &gt; 0 then
		idx = last - len + 1
	else
		for key in pairs(tbl) do
			if type(key) == 'number' and (idx == nil or
				key &lt; idx) then idx = key end
		end
		if idx == nil then return context_iterate(ctx, 3) end
		idx = idx - len
		if last &lt; idx then return context_iterate(ctx, 3) end
		len = last - idx + 1
	end
	ctx.params = copy_table_reduced(ctx.params, idx, len)
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|shifting|addend|pipe to
library.shifting = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local nshift = tonumber(ctx.pipe[1])
	if nshift == nil or nshift == 0 or math.floor(nshift) ~= nshift then
		error(modulename .. ', ‘shifting’: A non-zero integer number must be provided', 0) end
	local tbl = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'number' then tbl[key + nshift] = val
		else tbl[key] = val end
	end
	ctx.params = tbl
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|reversing_numeric_names|pipe to
library.reversing_numeric_names = function (ctx)
	local tbl, numerics, nmax = ctx.params, {}, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key], tbl[key] = val, nil
			if key &gt; nmax then nmax = key end
		end
	end
	for key, val in pairs(numerics) do tbl[nmax - key + 1] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|pivoting_numeric_names|pipe to
--[[
library.pivoting_numeric_names = function (ctx)
	local tbl = ctx.params
	local shift = #tbl + 1
	if shift &lt; 2 then return library.reversing_numeric_names(ctx) end
	local numerics = {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key] = val
			tbl[key] = nil
		end
	end
	for key, val in pairs(numerics) do tbl[shift - key] = val end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|mirroring_numeric_names|pipe to
--[[
library.mirroring_numeric_names = function (ctx)
	local nmax, nmin
	local tbl, numerics = ctx.params, {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key] = val
			tbl[key] = nil
			if nmax == nil then nmin, nmax = key, key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
		end
	end
	for key, val in pairs(numerics) do tbl[nmax + nmin - key] = val end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|swapping_numeric_names|pipe to
--[[
library.swapping_numeric_names = function (ctx)
	local tmp
	local tbl, cache, nsize = ctx.params, {}, 0
	for key in pairs(tbl) do
		if type(key) == 'number' then
			nsize = nsize + 1
			cache[nsize] = key
		end
	end
	table.sort(cache)
	for idx = math.floor(nsize / 2), 1, -1 do
		tmp = tbl[cache[idx] ]
		tbl[cache[idx] ] = tbl[cache[nsize - idx + 1] ]
		tbl[cache[nsize - idx + 1] ] = tmp
	end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|sorting_sequential_values|[criterion]|pipe to
library.sorting_sequential_values = function (ctx)
	local sortfn
	if ctx.pipe[1] ~= nil then
		sortfn = sortfunctions[ctx.pipe[1]:match'^%s*(.-)%s*$']
	end
	if sortfn then table.sort(ctx.params, sortfn)
	else table.sort(ctx.params) end -- i.e. either `false` or `nil`
	if sortfn == nil then return context_iterate(ctx, 1) end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|splicing|[add to position]|position|increment|
--            [number of elements to write]|...|pipe to
library.splicing = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp2, argc, pos, refp
	local opts, tbl = ctx.pipe, ctx.params
	local tmp1 = opts[1]
	if tmp1 ~= nil then
		tmp2 = tonumber(tmp1)
		if tmp2 == nil or math.floor(tmp2) ~= tmp2 then
			pos, argc, tmp2 = tonumber(opts[2]), 4,
				tmp1:match'^%s*(.*%S)'
			if tmp2 ~= nil then
				refp = position_references[tmp2]
				if refp == nil then error(modulename ..
					', ‘splicing’: ‘' .. tostring(tmp2) ..
					'’ is not a valid first argument', 0) end
			else refp = 0 end
		else pos, argc, refp = tmp2, 3, 0 end
	else pos, argc, refp = tonumber(opts[2]), 4, 0 end
	if pos == nil or math.floor(pos) ~= pos then error(modulename ..
		', ‘splicing’: The position must be an integer number', 0) end
	local len = tonumber(opts[argc - 1])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘splicing’: The increment must be an integer number', 0) end
	local insn = tonumber(opts[argc])
	if len == 0 and (insn == nil or insn &lt; 1) then error(modulename ..
		', ‘splicing’: When the increment is zero the number of elements to write cannot be zero', 0) end
	if refp == 2 then
		for _ in ipairs(tbl) do pos = pos + 1 end
		refp = 0
	end
	tmp1, tmp2 = nil, nil
	if refp ~= 0 or len ~= 0 then
		for key, val in pairs(tbl) do
			if type(key) == 'number' then
				if tmp1 == nil then tmp1, tmp2 = key, key
				elseif key &lt; tmp1 then tmp1 = key
				elseif key &gt; tmp2 then tmp2 = key end
			end
		end
	end
	if tmp2 == nil then len = 0
	elseif refp == 3 then pos = pos + tmp2
	elseif refp == 1 then pos = pos + tmp1 end
	if len &gt; 0 and pos + len &gt; tmp1 and pos &lt;= tmp2 then
		tbl = copy_table_expanded(tbl, pos, len)
	elseif len &lt; 0 and pos - len &gt; tmp1 and pos &lt;= tmp2 then
		tbl = copy_table_reduced(tbl, pos, -len)
	else tbl = copy_or_ref_table(tbl, tbl ~= ctx.oparams) end
	ctx.params = tbl
	if insn == nil or insn &lt; 0 or math.floor(insn) ~= insn then
		return context_iterate(ctx, argc)
	end
	tmp1 = argc - pos + 1
	for key = pos, pos + insn - 1 do tbl[key] = opts[key + tmp1] end
	return context_iterate(ctx, argc + insn + 1)
end


-- Syntax:  #invoke:params|imposing|name|value|pipe to
library.imposing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘imposing’: Missing parameter name to impose', 0) end
	ctx.params[get_parameter_name(ctx.pipe[1])] = ctx.pipe[2]
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|providing|name|value|pipe to
library.providing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘providing’: Missing parameter name to provide', 0) end
	local key = get_parameter_name(ctx.pipe[1])
	if ctx.params[key] == nil then ctx.params[key] = ctx.pipe[2] end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|reassigning|source|destination|[mode]|pipe to
library.reassigning = function (ctx)
	local opts, tbl = ctx.pipe, ctx.params
	if opts[1] == nil then error(modulename ..
		', ‘reassigning’: Missing source parameter', 0) end
	if opts[2] == nil then error(modulename ..
		', ‘reassigning’: Missing destination parameter', 0) end
	local mode, argc
	local src = get_parameter_name(opts[1])
	local val = tbl[src]
	if opts[3] ~= nil then mode = a_modes[opts[3]:match'^%s*(.-)%s*$'] end
	if mode == nil then mode, argc = 0, 3 else argc = 4 end
	if val == nil and mode &gt; 1 then return context_iterate(ctx, argc) end
	local dest = get_parameter_name(opts[2])
	local tmp = tbl[dest] == nil
	if mode % 2 == 0 or (mode == 7 and tmp) then tbl[src] = nil end
	if tmp or mode &lt; 4 then tbl[dest] = val end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|discarding|name|[how many]|pipe to
library.discarding = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘discarding’: Missing parameter name to discard', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil then
		ctx.params[get_parameter_name(ctx.pipe[1])] = nil
		return context_iterate(ctx, 2)
	end
	local key = tonumber(ctx.pipe[1])
	if key == nil or math.floor(key) ~= key then error(modulename ..
		', ‘discarding’: A range was provided, but the initial parameter name is not an integer number', 0) end
	if len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘discarding’: A range can only be an integer number greater than zero', 0) end
	for idx = key, key + len - 1 do ctx.params[idx] = nil end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|excluding_non-numeric_names|pipe to
library['excluding_non-numeric_names'] = function (ctx)
	local tmp = ctx.params
	for key, val in pairs(tmp) do
		if type(key) ~= 'number' then tmp[key] = nil end
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|excluding_numeric_names|pipe to
library.excluding_numeric_names = function (ctx)
	local tmp = ctx.params
	for key, val in pairs(tmp) do
		if type(key) == 'number' then tmp[key] = nil end
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|with_name_matching|target 1|[plain flag 1]|[or]
--            |[target 2]|[plain flag 2]|[or]|[...]|[target N]|[plain flag
--            N]|pipe to
library.with_name_matching = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp, ptn
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_name_matching')
	local tbl, newparams = ctx.params, {}
	for idx = 1, nptns do
		ptn = targets[idx]
		if ptn[3] then
			tmp = ptn[1]
			if tmp == '0' or tmp:find'^%-?[1-9]%d*$' ~= nil then
				tmp = tonumber(tmp)
			end
			newparams[tmp] = tbl[tmp]
		else
			for key, val in pairs(tbl) do
				if tostring(key):find(ptn[1], 1, ptn[2]) then
					newparams[key] = val
				end
			end
		end
	end
	ctx.params = newparams
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_name_not_matching|target 1|[plain flag 1]
--            |[and]|[target 2]|[plain flag 2]|[and]|[...]|[target N]|[plain
--            flag N]|pipe to
library.with_name_not_matching = function (ctx)
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_name_not_matching')
	local tbl = ctx.params
	if nptns == 1 and targets[1][3] then
		local tmp = targets[1][1]
		if tmp == '0' or tmp:find'^%-?[1-9]%d*$' ~= nil then
			tbl[tonumber(tmp)] = nil
		else tbl[tmp] = nil end
		return context_iterate(ctx, argc)
	end
	local yesmatch, ptn
	for key in pairs(tbl) do
		yesmatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if tostring(key) ~= ptn[1] then
					yesmatch = false
					break
				end
			elseif not tostring(key):find(ptn[1], 1, ptn[2]) then
				yesmatch = false
				break
			end
		end
		if yesmatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_value_matching|target 1|[plain flag 1]|[or]
--            |[target 2]|[plain flag 2]|[or]|[...]|[target N]|[plain flag
--            N]|pipe to
library.with_value_matching = function (ctx)
	local nomatch, ptn
	local tbl = ctx.params
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_value_matching')
	for key, val in pairs(tbl) do
		nomatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if val == ptn[1] then
					nomatch = false
					break
				end
			elseif val:find(ptn[1], 1, ptn[2]) then
				nomatch = false
				break
			end
		end
		if nomatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_value_not_matching|target 1|[plain flag 1]
--            |[and]|[target 2]|[plain flag 2]|[and]|[...]|[target N]|[plain
--            flag N]|pipe to
library.with_value_not_matching = function (ctx)
	local yesmatch, ptn
	local tbl = ctx.params
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_value_not_matching')
	for key, val in pairs(tbl) do
		yesmatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if val ~= ptn[1] then
					yesmatch = false
					break
				end
			elseif not val:find(ptn[1], 1, ptn[2]) then
				yesmatch = false
				break
			end
		end
		if yesmatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|keeping_at_most|number of parameters to pick|pipe to
library.keeping_at_most = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local len = tonumber(ctx.pipe[1])
	if len == nil or len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘keeping_at_most’: The number of parameters to keep must be an integer greater than zero', 0) end
	ctx.params = copy_table_maxn({}, ctx.params, len)
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|trimming_values|pipe to
library.trimming_values = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:match'^%s*(.-)%s*$' end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_to_lowercase|pipe to
library.mapping_to_lowercase = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:lower() end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_to_uppercase|pipe to
library.mapping_to_uppercase = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:upper() end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_by_calling|template name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            parameters]|[parameter 1]|[parameter 2]|[...]|[parameter N]|pipe to
library.mapping_by_calling = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘mapping_by_calling’: No template name was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.values_only, ctx.params)
	local model = { title = tname, args = margs }
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return ctx.frame:expandTemplate(model)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_invoking|module name|function name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.mapping_by_invoking = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘mapping_by_invoking’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘mapping_by_invoking’: No function name was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 2, mapping_styles.values_only, ctx.params)
	local model = { title = 'Module:' .. mname, args = margs }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘mapping_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return tostring(mfunc(ctx.frame:newChild(model)))
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_magic|parser function|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.mapping_by_magic = function (ctx)
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘mapping_by_magic’: No parser function was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.values_only, ctx.params)
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return ctx.frame:callParserFunction(magic, margs)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_replacing|target|replace|[count]|[plain
--            flag]|pipe to
library.mapping_by_replacing = function (ctx)
	local ptn, repl, nmax, flg, argc, die =
		load_replace_args(ctx.pipe, 'mapping_by_replacing')
	if die then return context_iterate(ctx, argc) end
	local tbl = ctx.params
	if flg == 3 then
		for key, val in pairs(tbl) do
			if val == ptn then tbl[key] = repl end
		end
	else
		if flg == 2 then
			-- Copied from Module:String's `str._escapePattern()`
			ptn = ptn:gsub('[%(%)%.%%%+%-%*%?%[%^%$%]]', '%%%0')
		end
		for key, val in pairs(tbl) do
			tbl[key] = val:gsub(ptn, repl, nmax)
		end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_mixing|mixing string|pipe to
library.mapping_by_mixing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘mapping_by_mixing’: No mixing string was provided', 0) end
	local tbl, mix = ctx.params, ctx.pipe[1]
	if mix == '$#' then
		for key in pairs(tbl) do tbl[key] = tostring(key) end
		return context_iterate(ctx, 2)
	end
	local skel, cnv, n_parts = parse_placeholder_string(mix)
	for key, val in pairs(tbl) do
		for idx = 2, n_parts, 2 do
			if skel[idx] then cnv[idx] = val
			else cnv[idx] = tostring(key) end
		end
		tbl[key] = table.concat(cnv)
	end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|mapping_to_names|pipe to
--[[
library.mapping_to_names = function (ctx)
	local tbl = ctx.params
	for key in pairs(tbl) do tbl[key] = tostring(key) end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|renaming_to_lowercase|pipe to
library.renaming_to_lowercase = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'string' then cache[key:lower()] = val else
		cache[key] = val end
	end
	ctx.params = cache
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|renaming_to_uppercase|pipe to
library.renaming_to_uppercase = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'string' then cache[key:upper()] = val else
		cache[key] = val end
	end
	ctx.params = cache
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|renaming_to_sequence|[sort order]|pipe to
library.renaming_to_sequence = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache, len
	local tbl = ctx.params
	local sortfn, argc, do_sort = load_sort_opt(ctx.pipe[1])
	if do_sort then
		local words, wl
		cache, words, len, wl = get_key_list_sorted(tbl, sortfn)
		for idx = 1, len do cache[idx] = tbl[cache[idx]] end
		for idx = 1, wl do cache[len + idx] = tbl[words[idx]] end
	else
		len, cache = 0, {}
		for _, val in pairs(tbl) do
			len = len + 1
			cache[len] = val
		end
	end
	ctx.params = cache
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_calling|template name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            parameters]|[parameter 1]|[parameter 2]|[...]|[parameter N]|pipe to
library.renaming_by_calling = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘renaming_by_calling’: No template name was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.names_only, ctx.params)
	local model = { title = tname, args = rargs }
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return ctx.frame:expandTemplate(model)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_invoking|module name|function
--            name|[call style]|[let/use]|[...]|[let/use]|[...]|[number of
--            additional arguments]|[argument 1]|[argument 2]|[...]|[argument
--            N]|pipe to
library.renaming_by_invoking = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘renaming_by_invoking’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘renaming_by_invoking’: No function name was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 2, mapping_styles.names_only, ctx.params)
	local model = { title = 'Module:' .. mname, args = rargs }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘renaming_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return tostring(mfunc(ctx.frame:newChild(model)))
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_magic|parser function|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.renaming_by_magic = function (ctx)
	local opts = ctx.pipe
	local magic
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘renaming_by_magic’: No parser function was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.names_only, ctx.params)
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return ctx.frame:callParserFunction(magic, rargs)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_replacing|target|replace|[count]|[plain
--            flag]|pipe to
library.renaming_by_replacing = function (ctx)
	local ptn, repl, nmax, flg, argc, die =
		load_replace_args(ctx.pipe, 'renaming_by_replacing')
	if die then return context_iterate(ctx, argc) end
	local tbl = ctx.params
	if flg == 3 then
		ptn = get_parameter_name(ptn)
		local val = tbl[ptn]
		if val ~= nil then
			tbl[ptn], tbl[get_parameter_name(repl)] = nil, val
		end
	else
		if flg == 2 then
			-- Copied from Module:String's `str._escapePattern()`
			ptn = ptn:gsub('[%(%)%.%%%+%-%*%?%[%^%$%]]', '%%%0')
		end
		local cache = {}
		for key, val in pairs(tbl) do
			steal_if_renamed(val, tbl, key, cache,
				tostring(key):gsub(ptn, repl, nmax))
		end
		for key, val in pairs(cache) do tbl[key] = val end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_mixing|mixing string|pipe to
library.renaming_by_mixing = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘renaming_by_mixing’: No mixing string was provided', 0) end
	local mix = ctx.pipe[1]:match'^%s*(.-)%s*$'
	local cache = {}
	if mix == '$@' then
		for _, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = val
		end
	else
		local skel, canvas, n_parts = parse_placeholder_string(mix)
		for key, val in pairs(ctx.params) do
			for idx = 2, n_parts, 2 do
				if skel[idx] then canvas[idx] = val
				else canvas[idx] = tostring(key) end
			end
			cache[get_parameter_name(table.concat(canvas))] = val
		end
	end
	ctx.params = cache
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|renaming_to_values|pipe to
--[[
library.renaming_to_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for _, val in pairs(ctx.params) do cache[val] = val end
	ctx.params = cache
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|grouping_by_calling|template
--            name|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.grouping_by_calling = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp, argc, tbl, mem = load_child_opts(ctx.pipe, 2, 0, ctx.params)
	local gargs = {}
	for key, val in pairs(tmp) do
		if type(key) == 'number' and key &lt; 1 then gargs[key - 1] = val
		else gargs[key] = val end
	end
	tmp = ctx.pipe[1]
	if tmp ~= nil then tmp = tmp:match'^%s*(.*%S)' end
	if tmp == nil then error(modulename ..
		', ‘grouping_by_calling’: No template name was provided', 0) end
	local model = { title = tmp }
	local groups = make_groups(tbl)
	for gid, group in pairs(groups) do
		for key, val in pairs(gargs) do group[key] = val end
		group[0], model.args = gid, group
		groups[gid] = ctx.frame:expandTemplate(model)
	end
	for key, val in pairs(mem) do groups[key] = val end
	ctx.params = groups
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|parsing|string to parse|[trim flag]|[iteration
--            delimiter setter]|[...]|[key-value delimiter setter]|[...]|pipe to
library.parsing = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘parsing’: No string to parse was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '|', '=')
	parse_parameter_string(ctx.params, opts[1], isep, iplain, psep, pplain,
		trimnamed, trimunnamed)
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|reinterpreting|parameter to reinterpret|[trim
--            flag]|[iteration delimiter setter]|[...]|[key-value delimiter
--            setter]|[...]|pipe to
library.reinterpreting = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘reinterpreting’: No parameter to reinterpret was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '|', '=')
	local tbl, tmp = ctx.params, get_parameter_name(opts[1])
	local str = tbl[tmp]
	if str ~= nil then
		tbl[tmp] = nil
		parse_parameter_string(tbl, str, isep, iplain, psep, pplain,
			trimnamed, trimunnamed)
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|evaluating|string to parse|[trim flag]|[iteration
--            delimiter setter]|[...]|[key-value delimiter setter]|[...]|pipe to
library.evaluating = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘evaluating’: No string to parse was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '!', ':')
	if opts[1]:match'^%s*(.*%S)' == nil then
		ctx.pipe = copy_or_ref_table(opts, opts ~= ctx.opipe)
		return context_iterate(ctx, argc)
	end
	local new_opts, cache = {}, {}
	local shift = parse_parameter_string(cache, opts[1], isep, iplain,
		psep, pplain, trimnamed, trimunnamed) - argc
	for key, val in pairs(opts) do
		if type(key) ~= 'number' or key &lt; 1 then new_opts[key] = val
		elseif key &gt;= argc then new_opts[key + shift] = val end
	end
	for key, val in pairs(cache) do new_opts[key] = val end
	ctx.pipe = new_opts
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mixing_names_and_values|mixing string|pipe to
library.mixing_names_and_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘mixing_names_and_values’: No mixing string was provided for parameter names', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘mixing_names_and_values’: No mixing string was provided for parameter values', 0) end
	local tmp
	local mix_k = ctx.pipe[1]:match'^%s*(.-)%s*$'
	local cache, mix_v = {}, ctx.pipe[2]
	if mix_k == '$@' and mix_v == '$@' then
		for _, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = val
		end
	elseif mix_k == '$@' and mix_v == '$#' then
		for key, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = tostring(key)
		end
	elseif mix_k == '$#' and mix_v == '$#' then
		for key in pairs(ctx.params) do cache[key] = tostring(key) end
	else
		local skel_k, cnv_k, n_parts_k = parse_placeholder_string(mix_k)
		local skel_v, cnv_v, n_parts_v = parse_placeholder_string(mix_v)
		for key, val in pairs(ctx.params) do
			tmp = tostring(key)
			for idx = 2, n_parts_k, 2 do
				if skel_k[idx] then cnv_k[idx] = val else cnv_k[idx] = tmp end
			end
			for idx = 2, n_parts_v, 2 do
				if skel_v[idx] then cnv_v[idx] = val else cnv_v[idx] = tmp end
			end
			cache[get_parameter_name(table.concat(cnv_k))] =
				table.concat(cnv_v)
		end
	end
	ctx.params = cache
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|swapping_names_and_values|pipe to
--[[
library.swapping_names_and_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do cache[val] = key end
	ctx.params = cache
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|combining|new parameter name|[sort
--            order]|[with/without flushed glue]|setting directives|...|pipe to
library.combining = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	return context_iterate(ctx, combine_parameters(
		ctx,
		function (key, val, kvs) return key .. kvs .. val end,
		'combining'
	) + 1)
end


-- Syntax:  #invoke:params|combining_values|new parameter name|[sort
--            order]|[with/without flushed glue]|setting directives|...|pipe to
library.combining_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	return context_iterate(ctx, combine_parameters(
		ctx,
		function (key, val, kvs) return val end,
		'combining_values'
	) + 1)
end


-- Syntax:  #invoke:params|combining_by_calling|template name|new parameter
--            name|pipe to
library.combining_by_calling = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tname = ctx.pipe[1]
	if tname ~= nil then tname = tname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_calling’: No template name was provided', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘combining_by_calling’: No parameter name was provided', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[2])] = ctx.frame:expandTemplate{
			title = tname,
			args = ctx.params
		}
	}
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|combining_by_invoking|module name|function name|new
--            parameter name|pipe to
library.combining_by_invoking = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local mname = ctx.pipe[1]
	if mname ~= nil then mname = mname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_invoking’: No module name was provided', 0) end
	local fname = ctx.pipe[2]
	if fname ~= nil then fname = fname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_invoking’: No function name was provided', 0) end
	if ctx.pipe[3] == nil then error(modulename ..
		', ‘combining_by_invoking’: No parameter name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = ctx.params }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘mapping_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[3])] =
			tostring(mfunc(ctx.frame:newChild(model)))
	}
	return context_iterate(ctx, 4)
end


-- Syntax:  #invoke:params|combining_by_magic|parser function|new parameter
--            name|pipe to
library.combining_by_magic = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local magic = ctx.pipe[1]
	if magic ~= nil then magic = magic:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_magic’: No parser function was provided', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘combining_by_magic’: No parameter name was provided', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[2])] =
			ctx.frame:callParserFunction(magic, ctx.params)
	}
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|snapshotting|[maximum number]|pipe to
library.snapshotting = function (ctx)
	return context_iterate(ctx, make_child(ctx, ctx.params, 'snapshotting'))
end


-- Syntax:  #invoke:params|remembering|[maximum number]|pipe to
library.remembering = function (ctx)
	return context_iterate(ctx, make_child(ctx, ctx.oparams, 'remembering'))
end


-- Syntax:  #invoke:params|entering_substack|[new]|pipe to
library.entering_substack = function (ctx)
	local tbl, ncurrparent = ctx.params, ctx.n_parents + 1
	if ctx.parents == nil then ctx.parents = { tbl }
	else ctx.parents[ncurrparent] = tbl end
	ctx.n_parents = ncurrparent
	if ctx.pipe[1] ~= nil and ctx.pipe[1]:match'^%s*new%s*$' then
		ctx.params = {}
		return context_iterate(ctx, 2)
	end
	local currsnap = ctx.n_children
	if currsnap &gt; 0 then
		ctx.params, ctx.children[currsnap], ctx.n_children =
			ctx.children[currsnap], nil, currsnap - 1
	else
		local newparams = {}
		for key, val in pairs(tbl) do newparams[key] = val end
		ctx.params = newparams
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|pulling|parameter name|pipe to
library.pulling = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘pulling’: No parameter to pull was provided', 0) end
	local tmp = ctx.n_parents
	local parent = tmp &lt; 1 and ctx.oparams or ctx.parents[tmp]
	tmp = get_parameter_name(opts[1])
	if parent[tmp] ~= nil then ctx.params[tmp] = parent[tmp] end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|recalling|parameter name|pipe to
library.recalling = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘recalling’: No parameter to recall was provided', 0) end
	local arg = get_parameter_name(opts[1])
	if ctx.oparams[arg] ~= nil then ctx.params[arg] = ctx.oparams[arg] end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|finding|parameter name|pipe to
--[[
library.finding = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘finding’: No parameter to find was provided', 0) end
	local arg = get_parameter_name(opts[1])
	local parent
	for idx = ctx.n_parents, 1, -1 do
		parent = ctx.parents[idx]
		if parent[arg] ~= nil then
			ctx.params[arg] = parent[arg]
			return context_iterate(ctx, 2)
		end
	end
	if ctx.oparams[arg] ~= nil then ctx.params[arg] = ctx.oparams[arg] end
	return context_iterate(ctx, 2)
end
]]--


-- Syntax:  #invoke:params|picking|number of parameters to pick|pipe to
--[[
library.picking = function (ctx)
	local len = tonumber(ctx.pipe[1])
	if len == nil or len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘picking’: The number of parameters to pick must be an integer greater than zero', 0) end
	if ctx.n_parents &lt; 1 then copy_table_maxn(ctx.params, ctx.oparams, len)
	else copy_table_maxn(ctx.params, ctx.parents[ctx.n_parents], len) end
	return context_iterate(ctx, 2)
end
]]--


-- Syntax:  #invoke:params|detaching_substack|pipe to
library.detaching_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘detaching_substack’: No substack has been created', 0) end
	local parent = ctx.parents[ncurrparent]
	for key in pairs(ctx.params) do parent[key] = nil end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|dropping_substack|pipe to
library.dropping_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘dropping_substack’: No substack has been created', 0) end
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents =
		ctx.parents[ncurrparent], nil, ncurrparent - 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|leaving_substack|pipe to
library.leaving_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘leaving_substack’: No substack has been created', 0) end
	local currsnap = ctx.n_children + 1
	if ctx.children == nil then ctx.children = { ctx.params }
	else ctx.children[currsnap] = ctx.params end
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents, ctx.n_children =
		ctx.parents[ncurrparent], nil, ncurrparent - 1, currsnap
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|merging_substack|pipe to
library.merging_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘merging_substack’: No substack has been created', 0) end
	local parent, child = ctx.parents[ncurrparent], ctx.params
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents = parent, nil,
		ncurrparent - 1
	for key, val in pairs(child) do parent[key] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|flushing|pipe to
library.flushing = function (ctx)
	if ctx.n_children &lt; 1 then error(modulename ..
		', ‘flushing’: There are no substacks to flush', 0) end
	local parent, currsnap = ctx.params, ctx.n_children
	for key, val in pairs(ctx.children[currsnap]) do parent[key] = val end
	ctx.children[currsnap], ctx.n_children = nil, currsnap - 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|setting_by_flushing|pipe to
library.setting_by_flushing = function (ctx)
	set_strings_from_substack(ctx, ctx, 'setting_by_flushing')
	return context_iterate(ctx, 1)
end


	--[[ Functions ]]--
	-----------------------------


-- Syntax:  #invoke:params|count
library.count = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local retval = 0
	for _ in ctx.iterfunc(ctx.params) do retval = retval + 1 end
	if ctx.subset == -1 then retval = retval - #ctx.params end
	ctx.text = retval
	return false
end


-- Syntax:  #invoke:args|concat_and_call|template name|[prepend 1]|[prepend 2]
--            |[...]|[item n]|[named item 1=value 1]|[...]|[named item n=value
--            n]|[...]
library.concat_and_call = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘concat_and_call’: No template name was provided', 0) end
	remove_numeric_keys(opts, 1, 1)
	ctx.text = ctx.frame:expandTemplate{
		title = tname,
		args = concat_params(ctx)
	}
	return false
end


-- Syntax:  #invoke:args|concat_and_invoke|module name|function name|[prepend
--            1]|[prepend 2]|[...]|[item n]|[named item 1=value 1]|[...]|[named
--            item n=value n]|[...]
library.concat_and_invoke = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘concat_and_invoke’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘concat_and_invoke’: No function name was provided', 0) end
	remove_numeric_keys(opts, 1, 2)
	local mfunc = require('Module:' .. mname)[fname]
	if mfunc == nil then error(modulename ..
		', ‘concat_and_invoke’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	ctx.text = mfunc(ctx.frame:newChild{
		title = 'Module:' .. mname,
		args = concat_params(ctx)
	})
	return false
end


-- Syntax:  #invoke:args|concat_and_magic|parser function|[prepend 1]|[prepend
--            2]|[...]|[item n]|[named item 1=value 1]|[...]|[named item n=
--            value n]|[...]
library.concat_and_magic = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘concat_and_magic’: No parser function was provided', 0) end
	remove_numeric_keys(opts, 1, 1)
	ctx.text = ctx.frame:callParserFunction(magic, concat_params(ctx))
	return false
end


-- Syntax:  #invoke:params|value_of|parameter name
library.value_of = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘value_of’: No parameter name was provided', 0) end
	local val
	local key = opts[1]:match'^%s*(.-)%s*$'
	if key == '0' or key:find'^%-?[1-9]%d*$' ~= nil then
		key = tonumber(key)
		val = ctx.params[key]
		-- No worries: #ctx.params is unused when the modifier is in
		-- first position (and therefore `ctx.params` is a metatable)
		if val ~= nil and (
			ctx.subset ~= -1 or key &gt; #ctx.params or key &lt; 1
		) and (
			ctx.subset ~= 1 or (key &lt;= #ctx.params and key &gt; 0)
		) then
			ctx.text = (ctx.header or '') .. val .. (ctx.footer or '')
		else ctx.text = ctx.ifngiven or '' end
	else
		val = ctx.params[key]
		if ctx.subset ~= 1 and val ~= nil then ctx.text = (ctx.header
			or '') .. val .. (ctx.footer or '')
		else ctx.text = ctx.ifngiven or '' end
	end
	return false
end


-- Syntax:  #invoke:params|list
library.list = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, kvs, pps = {}, 0, ctx.pairsep or '', ctx.itersep or ''
	flush_params(ctx, function (key, val)
		ret[nss + 1], ret[nss + 2], ret[nss + 3], ret[nss + 4], nss =
			pps, key, kvs, val, nss + 4
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 4)
	return false
end


-- Syntax:  #invoke:params|list_values
library.list_values = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	-- NOTE: `library.coins()` and `library.unique_coins()` rely on us
	local ret, nss, pps = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		ret[nss + 1], ret[nss + 2], nss = pps, val, nss + 2
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|list_maybe_with_names
library.list_maybe_with_names = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, kvs, pps = {}, 0, ctx.pairsep or '', ctx.itersep or ''
	mixed_flush_params(
		ctx,
		function (key, val)
			ret[nss + 1], ret[nss + 2], ret[nss + 3],
				ret[nss + 4], nss = pps, '', '', val, nss + 4
		end,
		function (key, val)
			ret[nss + 1], ret[nss + 2], ret[nss + 3],
				ret[nss + 4], nss = pps, key, kvs, val, nss + 4
		end
	)
	finalize_and_return_concatenated_list(ctx, ret, nss, 4)
	return false
end


-- Syntax:  #invoke:params|coins|[first coin = value 1]|[second coin = value
--            2]|[...]|[last coin = value N]
--[[
library.coins = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local opts, tbl = ctx.pipe, ctx.params
	for key, val in pairs(tbl) do tbl[key] = opts[get_parameter_name(val)] end
	return library.list_values(ctx)
end
]]--


-- Syntax:  #invoke:params|unique_coins|[first coin = value 1]|[second coin =
--            value 2]|[...]|[last coin = value N]
--[[
library.unique_coins = function (ctx)
	local tmp
	local opts, tbl = ctx.pipe, ctx.params
	for key, val in pairs(tbl) do
		tmp = get_parameter_name(val)
		tbl[key], opts[tmp] = opts[tmp], nil
	end
	return library.list_values(ctx)
end
]]


-- Syntax:  #invoke:params|for_each|wikitext
library.for_each = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, pps, txt = {}, 0, ctx.itersep or '', ctx.pipe[1] or ''
	local skel, cnv, n_parts = parse_placeholder_string(txt)
	flush_params(ctx, function (key, val)
		for idx = 2, n_parts, 2 do
			if skel[idx] then cnv[idx] = val
			else cnv[idx] = tostring(key) end
		end
		ret[nss + 1], nss = pps, nss + 2
		ret[nss] = table.concat(cnv)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each|template name|[append 1]|[append 2]
--            |[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘call_for_each’: No template name was provided', 0) end
	local model = { title = tname, args = opts }
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	table.insert(opts, 1, true)
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = ctx.frame:expandTemplate(model)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|invoke_for_each|module name|module function|[append
--            1]|[append 2]|[...]|[append n]|[named param 1=value 1]|[...]
--            |[named param n=value n]|[...]
library.invoke_for_each = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘invoke_for_each’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘invoke_for_each’: No function name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = opts }
	local mfunc = require(model.title)[fname]
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = mfunc(ctx.frame:newChild(model))
		
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|magic_for_each|parser function|[append 1]|[append 2]
--            |[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.magic_for_each = function (ctx)
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘magic_for_each’: No parser function was provided', 0) end
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	table.insert(opts, 1, true)
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = ctx.frame:callParserFunction(magic, opts)

	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each_value|template name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each_value = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘call_for_each_value’: No template name was provided', 0) end
	local model = { title = tname, args = opts }
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = ctx.frame:expandTemplate(model)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|invoke_for_each_value|module name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.invoke_for_each_value = function (ctx)
	local opts = ctx.pipe
	local mname, fname
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘invoke_for_each_value’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘invoke_for_each_value’: No function name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = opts }
	local mfunc = require(model.title)[fname]
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	remove_numeric_keys(opts, 1, 1)
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = mfunc(ctx.frame:newChild(model))
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|magic_for_each_value|parser function|[append 1]
--            |[append 2]|[...]|[append n]|[named param 1=value 1]|[...]|[named
--            param n=value n]|[...]
library.magic_for_each_value = function (ctx)
	local opts = ctx.pipe
	local magic
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘magic_for_each_value’: No parser function was provided', 0) end
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = ctx.frame:callParserFunction(magic, opts)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each_group|template name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each_group = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local tmp
	if ctx.pipe[1] ~= nil then tmp = ctx.pipe[1]:match'^%s*(.*%S)' end
	if tmp == nil then error(modulename ..
		', ‘call_for_each_group’: No template name was provided', 0) end
	local model = { title = tmp }
	local opts, ret, nss, ccs = {}, {}, 0, ctx.itersep or ''
	for key, val in pairs(ctx.pipe) do
		if type(key) == 'number' then opts[key - 1] = val
		else opts[key] = val end
	end
	ctx.pipe = opts
	ctx.params = make_groups(ctx.params)
	flush_params(ctx, function (gid, group)
		for key, val in pairs(opts) do group[key] = val end
		group[0], model.args, ret[nss + 1], nss = gid, group, ccs,
			nss + 2
		ret[nss] = ctx.frame:expandTemplate(model)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


	--[[ First-position-only modifiers ]]--
	---------------------------------------


-- Syntax:  #invoke:params|new|pipe to
static_iface.new = function (child_frame)
	local ctx = context_new(child_frame)
	ctx.pipe = copy_or_ref_table(ctx.opipe, false)
	ctx.params = {}
	main_loop(ctx, context_iterate(ctx, 1))
	return ctx.text
end


	--[[ First-position-only functions ]]--
	---------------------------------------


-- Syntax:  #invoke:params|self
static_iface.self = function (frame)
	return frame:getParent():getTitle()
end


	--[[ Public metatable of functions ]]--
	---------------------------------------


return setmetatable({}, {
	__index = function (_, query)
		local fname = query:match'^%s*(.*%S)'
		if fname == nil then error(modulename ..
			': You must specify a function to call', 0) end
		local func = static_iface[fname]
		if func ~= nil then return func end
		func = library[fname]
		if func == nil then error(modulename ..
			': The function ‘' .. fname .. '’ does not exist', 0) end
		return function (child_frame)
			local ctx = context_new(child_frame)
			ctx.pipe = copy_or_ref_table(ctx.opipe, refpipe[fname])
			ctx.params = copy_or_ref_table(ctx.oparams, refparams[fname])
			main_loop(ctx, func)
			return ctx.text
		end
	end
})</text>
      <sha1>9ko1dij40e2ajwgg59fvqm72uoivi2f</sha1>
    </revision>
  </page>
  <page>
    <title>African Arthropods/Crabroninae</title>
    <ns>0</ns>
    <id>316633</id>
    <revision>
      <id>2819174</id>
      <parentid>2819128</parentid>
      <timestamp>2026-07-23T19:27:33Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Genus Liris Fabricius, 1804 */</comment>
      <origin>2819174</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6531" sha1="b3mq3trdi5rhaev2rybh1xtp2s0bwyp" xml:space="preserve">All [[w:Crabronidae|crabronid]] species known in the [[w:Afrotropical realm|Afrotropics]] are of the subfamily Crabroninae. Several species of the subfamily [[w:Dinetinae|Dinetinae]] are found in Egypt, Tunisia and Algeria ([[w:Palearctic realm|Palearctic]]).&lt;ref name=waspweb&gt;van Noort, S. 2024. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/Crabronidae/Classification/index.htm (accessed on 5 February 2025)&lt;/ref&gt;

==South African wasps of Tribe Crabronini==
&lt;gallery mode=packed heights=200&gt;
Dasyproctus iN 27385880 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Dasyproctus|''Dasyproctus'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Larrini==
&lt;gallery mode=packed heights=200&gt;
Dicranorhina kohli.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps (''[[w:Dicranorhina|Dicranorhina kohli]]'')
Liris haemorrhoidalis iN 47526569 04.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Liris on Crassula iN 42678436 01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Liris  |''Liris haemorrhoidalis'']])
Tachysphex specie edit1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' sp.]])
Tachysphex iN 250449986 2024 10 09 7305.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']])
Tachytes iN 188902572 1964.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Tachytes |''Tachytes'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Miscophini==
&lt;gallery mode=packed heights=200&gt;
Miscophus kriechbaumeri 2026-03-05.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Miscophus |''Miscophus kriechbaumeri'']])
Paranysson iN 199673489 1.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Paranysson|''Paranysson'' sp.]])
&lt;/gallery&gt;

==South African wasps of Tribe Oxybelini==
&lt;gallery mode=packed heights=200&gt;
Oxybelus iN 250449990 2024 10 09 - 02.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
Oxybelus iN 250450006 2024-10-09 07.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Oxybelus|''Oxybelus'' sp.]])
&lt;/gallery&gt;


==South African wasps of Tribe Palarini==
&lt;gallery mode=packed heights=200&gt;
Palarus_Bee_Pirate_iN_144133368_2022-12-01_01.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Palarus|''Palarus pentheri'']])
&lt;/gallery&gt;


==South African wasps of Tribe Trypoxylini==
&lt;gallery mode=packed heights=200&gt;
Pison iN 144131685 2022-11-30 03.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Pison|''Pison'' sp.]])
Trypoxylon iN 99063113 a.jpg|[[w:Crabronidae|Crabronidae]] - sand wasps ([[w:Trypoxylon|''Trypoxylon'' sp.]])
&lt;/gallery&gt;

== South African wasp species of Subtribe Larrina &lt;small&gt;Latreille, 1810&lt;/small&gt; (Larrini)==

===Genus ''Dicranorhina'' Shuckard, 1840===
*''Dicranorhina kohli'' (Brauns, 1899) (South Africa)
&lt;gallery mode=packed heights=200&gt;
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Dicranorhina kohli 2023 07 25 iN 176115975 b.jpg
Dicranorhina kohli.jpg
&lt;/gallery&gt;

=== Genus ''Larra'' Fabricius, 1793 ===
Separation of ''Larra'' from ''Liris'' is difficult (Bohart and Menke, 1976, p.235). A good view of the side of the propodeum is necessary to see the degree of punctuation (smooth in ''Liris'', distinct punctures in ''Larra''). Two South African species (''extrema'' and ''outeniqua'') have ferruginous or blackish-red legs; the third (nigripes) has a red gaster with a black tip:
*''Larra extrema'' Dahlbom, 1845 (South Africa)
 &quot;Abdomen totum nigrum; os, antenne et pedes ex parte, piceo-rufa&quot; (The abdomen is entirely black; the mouth, antennae and feet are partly blackish-red.)&lt;ref&gt;Dahlbom 1845 Hymenoptera Europaea praecipue borealia; ... http://www.biodiversitylibrary.org/page/15796684&lt;/ref&gt;
*''Larra nigripes'' (Fabricius, 1793) (South Africa)
 &quot;Black, gaster red, segment 5 somewhat blackish, 6 black. Wings not very dark, pale brownish with yellowish tinge. Gaster impunctate. Interocular distance on vertex = 14.5, at base of clypeus = 23.5, total width of head = 41, antennal segments 3-5 = 8.5, 5.5, 6.&lt;ref&gt;van der Vecht, 1961a. Hymenoptera Sphecoidea Fabriciana. Zoologische Verhandelingen 48:18. http://www.repository.naturalis.nl/document/149001&lt;/ref&gt;
*''Larra outeniqua'' Arnold, 1923 (South Africa)
 &quot;Black; mandibles, lower margins of the scapes, all the femora, tibiae and tarsi, and extreme apex of last sternite, ferruginous. Wings pale smoky, the veins black.&quot;&lt;ref&gt;Arnold, G.  1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:224. https://journals.co.za/toc/nfi.annalstm/9/4&lt;/ref&gt;

**''Larra zarudniana'' Gussakovskij, 1933 is from Asia: 
 &quot;♀. Holotype: ♀, Iran: Kerman: Bazman (ZIN). – R. Bohart and Menke, 1976:238 (in checklist of world Sphecidae); Jahantigh, Rakhshani, Mokhtari, and Ramroodi, 2017:65 (Iran: known from Sistan-o Baluchestan Province); Ghaderipour, Khayrandish, Madjdzadeh, Ebrahimi, and Schmid-Egger, 2020:118 (Iran: Kerman).&quot; &lt;ref&gt;Paulawski (2024) Larra http://researcharchive.calacademy.org/research/entomology/Entomology_Resources/Hymenoptera/sphecidae/genera/Larra.pdf&lt;/ref&gt;

=== Genus ''Liris'' Fabricius, 1804 ===
&lt;gallery mode=packed heights=200&gt;
Liris inaturalist 197329080.jpg|''Liris'' wasps hunt crickets
&lt;/gallery&gt;
*''Liris alaris felinus'' (Arnold, 1923) (South Africa)

*''Liris antaka transvaalensis'' (Cameron, 1910) (South Africa, Ethiopia)

*''Liris aterrimus'' (Smith, 1856) (South Africa)

*''Liris atropos'' Gribodo, 1894 (Mozambique, South Africa, Zimbabwe)

*''Liris bembesianus'' (Bischoff, 1913) (Namibia, Zimbabwe, [South Africa?])

*''Liris ciliatus'' (Smith, 1856) (South Africa)

*''Liris denticulatus'' (Turner, 1920) (South Africa)

*''Liris diabolicus'' (Smith, 1873) (Kenya, South Africa, Democratic Republic of Congo)

*''Liris haemorrhoidalis'' (Fabricius, 1804) (Old World)
&lt;gallery mode=packed heights=200&gt;
&lt;gallery mode=packed heights=200&gt;
Liris on Crassula iN 42678436 03.jpg|''Liris haemorrhoidalis''
&lt;/gallery&gt;
*''Liris memnonia'' (Smith, 1856) (South Africa, Afrotropics, Palearctic)

*''Liris niger namana'' (Bischoff, 1913) (South Africa)

*''Liris nugax'' (Kohl, 1894) (South Africa, Ethiopia, Malawi)

*''Liris rufoscapus'' (Cameron, 1905) (South Africa)

*''Liris sepulchralis'' (Gerstaecker in Peters, 1858) (South Africa, Afrotropics)

*''Liris solstitialis'' (Smith, 1856) (Madagascar, South Africa, Tanzania)

*''Liris subfasciatus'' (Walker, 1871) (South Africa, Afrotropics)

*''Liris transvaalensis'' (Cameron, 1910) (South Africa)

==References==
{{reflist}}

{{BookCat}}</text>
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[[File:Sorbus torminalis Trunk and canopy.jpg|thumb|310px|The intracanopy of a Wild Service Tree, i.e. &lt;small&gt;''Torminalis glaberrima'' (Gand.) Sennikov &amp; Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).&lt;/small&gt;]]&lt;br /&gt;
 
Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,&lt;small&gt;&lt;sup&gt;[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist (verify)]&lt;/sup&gt;&lt;/small&gt; to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://se.wikimedia.org/w/index.php?title=Special:Användare&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt; and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt;

So far (July 23, 2026), I've made '''just over 398,000 edits''' to 153 of the Wikimedia sister projects&amp;nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found '''[[meta:Special:CentralAuth/Tommy Kronkvist|here]]'''.

Swedish is my mother tongue&amp;nbsp;– even though I was born in Finland&amp;nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden.

I'm only the fourth generation named &quot;Kronkvist&quot;. My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning &quot;Matthew's Farm&quot;, dating back to at least 1637.
{{Clear}}
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    <title>WikiJournal Preprints/Mental health in Sri Lanka</title>
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      <text bytes="93764" sha1="ls2cf6qei0m3jn9i6mh17nzj1fl9j4c" xml:space="preserve">{{Article info
| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| last1 = Azeez
| orcid1 = 0009-0007-9202-4614
| first1 = Aaqib
| last2 = 
| first2 = 
| last3 = 
| first3 = 
| last4 = 
| first4 = &lt;!-- up to 9 authors can be added in this above format --&gt;
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| affiliation1 = Old Dominion University
| correspondence1 = aaqib.azeez@yahoo.com
| affiliations = institutes / affiliations
| correspondence = email@address.com
| keywords = &lt;!-- up to 6 keywords --&gt;
| license = &lt;!-- default is CC-BY --&gt;
| abstract = Mental health issues continue to be a significant problem in Sri Lanka, with 2022 suicide rates in the country reporting 15 suicides per 100,000 people, above the global average of 9 suicides per 100,000 people. The barriers to mental healthcare on the island are multi-faceted and are best understood with historical context. This narrative review covers the historical developments of mental healthcare,  mental health impacts of historical events within the last 100 years, current challenges affecting mental health outcomes, the role of the island's major religions in mental health and mental healthcare, and recommendations for improving future mental healthcare. The author uses peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports to support clinical and historical claims, though non-peer-reviewed sources were used to contextualize historical and non-clinical claims. The narrative review concludes that outdated legislation, impacts from recent conflicts or disasters, stigma surrounding mental health, and economic vulnerability contribute to mental health issues and the inefficiency of mental healthcare services. The author recommends updating legal frameworks, expanding services, and raising awareness to mitigate social stigma.
}}

== Introduction ==
Mental health continues to be a critically relevant topic as the island nation has experienced decades of [[w:Black_July|violent ethnic conflict]], terrorist attacks, alleged war crimes, and economic disruptions. Sri Lanka continues to recover from a [[w:Sri_Lankan_economic_crisis_(2019–2024)|severe economic crisis (2019 - 2024)]], a [[w:Sri_Lankan_civil_war|nearly 30-year civil war ending in 2009]], a [[w:2019_Sri_Lanka_Easter_bombings|2019 terrorist attack]], and the [[w:2004_Boxing_Day_tsunami|2004 Boxing Day tsunami]]. The exact effect these major events have had on mental health in the country is &quot;unknown&quot;, but the statistics remain concerning despite a declining trend in the overall suicide rate.

Suicide rates in the country during the mid-1990s were the second-highest in the world, with ingesting toxic products being the main suicide method. Despite the decline in suicide numbers since then—possibly attributed to Sri Lanka's ban on toxic products—evidence from a 2023 study reports an upward trend in suicide through hanging from 2016 to 2021—independent of the [[w:COVID-19_pandemic_in_Sri_Lanka|COVID-19 pandemic]]. Several risk factors for suicide, such as poverty and economic instability, are still prevalent and even increasing in the country&lt;ref&gt;{{Cite journal|last=Rajapakse|first=Thilini|last2=Silva|first2=Tharuka|last3=Hettiarachchi|first3=Nirosha Madhuwanthi|last4=Gunnell|first4=David|last5=Metcalfe|first5=Chris|last6=Spittal|first6=Matthew J.|last7=Knipe|first7=Duleeka|date=2023-01-19|title=The Impact of the COVID-19 Pandemic and Lockdowns on Self-Poisoning and Suicide in Sri Lanka: An Interrupted Time Series Analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9914278/|journal=International Journal of Environmental Research and Public Health|volume=20|issue=3|pages=1833|doi=10.3390/ijerph20031833|issn=1660-4601|pmc=9914278|pmid=36767200}}&lt;/ref&gt;.

== Methods ==
A narrative review was conducted on mental health in Sri Lanka. Sources used included peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports. These sources were found on Google Scholar, PubMed/PMC, Sri Lankan journals, and official Sri Lankan governmental websites showing relevant statistics/reports. Keywords used to conduct searches included, but were not limited to: &quot;Sri Lanka mental health&quot;, &quot;Sri Lanka civil war trauma&quot;, &quot;Sri Lanka suicide&quot;, &quot;Sri Lanka mental health ordinances&quot;, &quot;Sri Lanka religion and mental health&quot;, &quot;Sri Lanka public mental healthcare&quot;, and &quot;Sri Lanka poverty/economic crisis mental health impact.&quot; Literature included were relevant to the topic (Sri Lanka, South Asian mental health law, suicide, public mental health, conflict/disaster trauma, or cultural/religious practice), had full text available, and were in the English language. Non-peer-reviewed sources were primarily used to explain historical claims or contextualize non-clinical claims. 

==Historical Development of Mental Health Services==

Records attest to the care of the mentally ill through established hospitals in the island since the 4th century.&lt;ref name=&quot;:17&quot; /&gt; Prior to the incarceration of the mentally ill by the European colonizing forces, the mentally ill were regarded as ''Pissowetitch'', or people who had &quot;the spirit of the Gods within him&quot; and &quot;whatsoever he pronounceth, is looked upon as spoken by God himself, and the people will speak to him, as if it were the very person of God&quot;&lt;ref&gt;{{Cite web|url=https://www.gutenberg.org/files/14346/14346-h/14346-h.htm|title=An Historical Relation Of the Island Ceylon, in the East-Indies: Together, With an Account of the Detaining in Captivity the Author and divers other Englishmen now Living there, and of the Author’s Miraculous Escape.|last=Knox|first=Robert|website=www.gutenberg.org|language=en-us|access-date=2026-06-29}}&lt;/ref&gt;. With this religious understanding, Lucien de Alwis reasoned that the mentally ill in Sri Lanka were &quot;placed... at a higher social status than the mentally ill in the Western world&quot;, with this understanding correlating with the unsurprising absence of evidence of any &quot;large scale segregation[s] of [the] mentally ill from society&quot;&lt;ref name=&quot;:17&quot; /&gt;. 

In the 1800s, established care for mental health began shifting primarily from indigenous practices, mainly derived from [[w:Ayurveda|Ayurveda medicine]], [[w:Siddha_medicine|Siddha medicine]], and [[w:Unani_medicine|Unani medicine]], to a Western model by the British&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;Gambheera, H. (2011). [https://www.saarcpsychiatry.com/viewText?chapter=c6 The evolution of psychiatric services in Sri Lanka]. South Asian Journal of Psychiatry, 2(1), 25–27.&lt;/ref&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book|url=https://doi.org/10.1007/978-981-96-8078-8_7|title=Social Psychiatry in Sri Lanka|last=Baminiwatta|first=Anuradha|last2=Williams|first2=Shehan|date=2025|publisher=Springer Nature|isbn=978-981-96-8078-8|editor-last=Arafat|editor-first=S. M. Yasir|location=Singapore|pages=141–158|language=en|doi=10.1007/978-981-96-8078-8_7|editor-last2=Singh|editor-first2=Amit|editor-last3=Kar|editor-first3=Sujita Kumar}}&lt;/ref&gt;. 

=== Adoption of a Western-based mental healthcare model and ordinances ===
In 1839, [[w:James_Alexander_Stewart-Mackenzie|James Alexander Stewart-Mackenzie]], the 7th Governor of British Ceylon, released the Lunacy Ordinance, authorizing municipal authorities to create lunatic asylums for the mentally ill&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=6&amp;Itemid=125&amp;lang=en|title=History - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-10}}&lt;/ref&gt;. The ordinance was concerned with the legal frameworks of detaining individuals considered dangerous to others or individuals falsely presenting themselves as mentally ill, and not on medical treatments to alleviate the conditions of detained individuals. UK psychiatrist [[w:Edward_Mapother|Edward Mapother]] critiqued the ordinance during his 1937 inspection of British Ceylon's mental health institutions in a series of reports titled ''A Disgrace to a Civilised Community'', remarking that the ordinance &quot;[did] not seem to have contemplated treatment as a contingency to be considered&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Permeable walls: historical perspectives on hospital and asylum visiting|date=2009|publisher=Rodopi|isbn=978-90-420-2599-8|editor-last=Mooney|editor-first=Graham|series=Clio medica|location=Amsterdam New York, NY|editor-last2=Reinarz|editor-first2=Jonathan}}&lt;/ref&gt;.

The 1839 Ordinance was repealed and replaced by the 1840 Ordinance, which removed two requirements from the previous Ordinance: the requirement for official medical diagnoses of the mentally ill and the mandate to maintain adequate staff-to-patient ratios within lunatic asylums&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. de|last2=Seneviratne|first2=V. L.|last3=Mendis|first3=T. S. S.|last4=Abhayanayaka|first4=C.|date=2024-12-31|title=The development of laws related to the disposal of forensic patients in Sri Lanka: A historical review|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v15i2.8569|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=15|issue=2|doi=10.4038/sljpsyc.v15i2.8569|issn=2012-6883}}&lt;/ref&gt;.

In 1873, a third Ordinance was released. It included linguistic changes, where the term, &quot;insane&quot;, was replaced with &quot;of unsound mind&quot;. The Ordinance also gave more power to medical professionals in determining insanity diagnoses, and more power to detainees in appealing their commitment to the mental asylum. Despite the increased granted authority, the legal frameworks behind the detainment of the criminally insane were left identical to previous ordinances&lt;ref name=&quot;:3&quot; /&gt;.

=== Development of mental asylums ===
At the time the 1839 ordinance was released, mentally ill patients were placed either in prisons throughout the country or leprosy hospitals, such as the [[w:Hendala_Leprosy_Hospital|Hendala Leprosy Hospital]] in the Gampaha district&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;. After the creation of the first mental asylum in Borella in 1846, patients from the Hendala Leprosy Hospital were transferred to Borella. Overcrowding soon became an issue, which led to patients being sent to prisons. [[File:Edward Mapother.jpg|thumb|A portrait taken of Edward Mapother during his time working at [[w:Maudsley_Hospital|Maudsley Hospital]] in London.
]]

As medical institutions were being made to house the mentally ill, another mental asylum was created in the [[w:Cinnamon_Gardens|Cinnamon Gardens]] area of Colombo in 1884, though this mental asylum faced overcrowding issues in just one year&lt;ref name=&quot;:0&quot; /&gt;. Treatment in these asylums was limited to occupational and protection therapy, failing to provide treatment for the root causes.

In 1926, the Angoda Mental Hospital was established, marginally alleviating the severe overcrowding issues that were plaguing the preceding mental asylums. Despite the addition of 1,700 beds to the facility, treatment was still vastly limited and the patients were left in significantly poor conditions. 

=== Edward Mapother and his 1937 inspection of British Ceylon ===
Edward Mapother was born in Dublin, Ireland, on July 12, 1881 and moved to London when he was 7 years old&lt;ref&gt;{{Cite book|title=Madness to mental illness: a history of the Royal College of Psychiatrists|last=Bewley|first=Thomas|date=2008|publisher=RCPsych Publications ; Distributed in North America by Balogh International|isbn=978-1-904671-35-0|location=London : [S.l.]}}&lt;/ref&gt;. Mapother attained his M.D. in 1908. While Mapother was the Medical Superintendent of Maudsley Hospital in London, England, he was invited to inspect British Ceylon's mental health institutions by Dr S. T. Gunasekara, the first Medical Director of British Ceylon&lt;ref name=&quot;:1&quot; /&gt;.

In Mapother's visit, he commented that the Angoda Mental Hospital had the atmosphere of &quot;a prison that is neglected and dilapidated&quot;&lt;ref name=&quot;:1&quot; /&gt;. Overcrowding was still a major issue, with the institute hosting 3,000 patients—more than double the intended capacity. Patients were sleeping on mats and were clearly out of reach of adequate treatment. Mapother also noted that only 4% of public health expenditure in the country was being set for hospitals, drawing a stark comparison to London's 25%&lt;ref name=&quot;:1&quot; /&gt;. Mapother offered a vivid and grim account of the hospital in his reports:
&lt;blockquote&gt;
The floor, roof and walls of each cell consist alike of drab cement without any attempt at colouring or decoration. High up in one wall is a small window with stout iron bars. In the floor is a large hole into which the patient may pass his motion and urine. These cells are incompletely divided from one another by a partition which does not reach the roof so that the noise and stink from any one cell may reach at least all the others of the same row. Into these empty cells I was informed that the most noisy and troublesome patients in the hospital; were turned at night completely naked. The doors of the cell contain no observation window, and considering the violent character of many of these patients there is every ground for believing that the doors are rarely opened in the night by the solitary attendant on duty. It needs little imagination to picture the suffering of any patient in an early stage of bodily illness passing a night under such conditions, a situation which must frequently arise. I am told that the noise proceeding from this building is like that on a bad night in a menagerie&lt;ref name=&quot;:0&quot; /&gt;.&lt;/blockquote&gt;Mapother proposed a series of reinforcements to the legal, institutional, and medical frameworks of mental health care in British Ceylon. This included the decentralization of the psychiatric services, a reworking of the Lunacy Ordinance to incorporate treatment into the legal framework, and the establishment of a separate service of medical professionals dedicated to psychiatry. Mapother's recommendations led to several of the best local medical professionals to be sent to London for extensive training in psychiatry, while nurses from England were sent to British Ceylon to supervise hospital operations and train local staff&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;.

On August 25, 1938, the Executive Committee of Health approved the strategies proposed by Mapother, though the Government was unable to fully implement all of Mapother's interventions due to the 'heavy cost'. In fact, the Government decided to forego one of his proposals at the behest of the &quot;Visiting Committee&quot;, a committee that was tasked to &quot;meet at the hospital, carry out inspections, and make recommendations&quot; to the Executive Committee of Health&lt;ref name=&quot;:1&quot; /&gt;. The Government believed that deficiencies in their mental healthcare system could prove to be &quot;costly&quot; for their reputation, which enraged Mapother. Mapother intended to contact the Secretary of State regarding the &quot;distortion&quot; of his plans, but was interrupted by events preceding [[w:World_War_II|World War II]]&lt;ref name=&quot;:1&quot; /&gt;. Mapother passed away on March 20, 1940, without materializing his follow-up plans.

=== Post-Mapother developments and further innovations ===
[[File:Sri Lanka districts Colombo.svg|thumb|A map of Sri Lanka highlighting the Colombo District, where the capital is located. 
|right|250px]]Mapother's insights on the mental healthcare structure in British Ceylon proved to be the catalyst of significant renovations. In 1939, the first outpatient clinic was established in the [[w:National_Hospital_of_Sri_Lanka|National Hospital of Sri Lanka]] in Colombo. The first trained Ceylonese psychiatrists began practice in the 1940s, leading to the establishment of the first neuropsychiatric clinic in Colombo in 1943. Treatments for the mentally ill improved dramatically, as [[w:insulin_shock_therapy|insulin shock therapy]] and [[w:Electroconvulsive_therapy|cardiazol convulsive therapy]] were utilized&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Kathriarachchi|first=Samudra T.|last2=Seneviratne|first2=V. Lakmi|last3=Amarakoon|first3=Luckshika|date=2019-06|title=Development of Mental Health Care in Sri Lanka: Lessons Learned|url=https://journals.lww.com/tpsy/fulltext/2019/33020/development_of_mental_health_care_in_sri_lanka_.1.aspx|journal=Taiwanese Journal of Psychiatry|language=en-US|volume=33|issue=2|pages=55|doi=10.4103/TPSY.TPSY_15_19|issn=1028-3684}}&lt;/ref&gt;. Mapother's advocation for the decentralization of services were further honored through the 1947 establishment of a first child guidance clinic in Colombo General Hospital&lt;ref name=&quot;:0&quot; /&gt;. 

In 1948, British Ceylon was granted independence after the [[w:Sri_Lankan_independence_movement|Sri Lankan independence movement]]. Changes in the mental healthcare structure were not immediate following independence, but rapid expansions of mental healthcare services were continuing to actualize.  

The following decades saw positive institutional developments, such as the creation of a second hospital in [[w:Mulleriyawa|Mulleriyawa]] in 1957, and the creation of a psychiatric inpatient unit in Colombo General Hospital in 1967—effectively granting the city of Colombo the luxury of hosting the top psychiatric care in the country&lt;ref name=&quot;:5&quot;&gt;{{Cite book|url=http://link.springer.com/10.1007/978-1-4899-7999-5_4|title=Mental Health System Development in Sri Lanka|last=Minas|first=Harry|last2=Mendis|first2=Jayan|last3=Hall|first3=Teresa|date=2017|publisher=Springer US|isbn=978-1-4899-7997-1|editor-last=Minas|editor-first=Harry|location=Boston, MA|pages=59–77|language=en|doi=10.1007/978-1-4899-7999-5_4|editor-last2=Lewis|editor-first2=Milton}}&lt;/ref&gt;. The 1950s was also the start of psychopharmacological innovations, with the introduction of [[w:Lithium_(medication)|lithium]] and long-acting injectable antipsychotics ([[w:Depot_injection|depot]] [[w:Antipsychotic|neuroleptics]]) in the succeeding years&lt;ref name=&quot;:4&quot; /&gt;. Additionally, the number of public psychiatrist positions increased by 400% from 1953 to 1967&lt;ref name=&quot;:5&quot; /&gt;.

After 1960, mental health services were expanded from beyond the capital to other cities in the country&lt;ref name=&quot;:2&quot; /&gt;.

In 1980, the [[w:Postgraduate_Institute_of_Medicine|Postgraduate Institute of Medicine]] initiated a program where students would enroll in a 5-year medical course and attain an MD in psychiatry, curbing the need for Sri Lankan medical students to be sent abroad to complete their training. Many of the medical students sent abroad for training never returned to Sri Lanka to practice, resulting in a &quot;1:500,000 to 1000,000&quot; ratio of psychiatrists to patients on &quot;most occasions&quot;&lt;ref name=&quot;:0&quot; /&gt;.

=== Mental Disease Ordinance of 1956 ===
In 1956, the 1873 Ordinance was revised a second time. The Mental Disease Ordinance of 1956 featured another linguistic development, as &quot;lunacy&quot; was replaced with &quot;mental disease&quot;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}&lt;/ref&gt;. The Ordinance paved way for community-based services to be delivered to patients closer to their residences, rather than strictly allocating services to just hospitals. This led to the creation of a [[w:WHO|WHO]]-backed community clinic near the [[w:University_of_Colombo|University of Colombo]] in the 1970s, where the focus was to eventually ease patients in the Angoda Mental Hospital back into the general population&lt;ref name=&quot;:5&quot; /&gt;.

=== Developments from the 1990s ===
The 1990s and onwards saw further positive developments in framing the mental healthcare system, including the establishment of the [https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=featured&amp;Itemid=101&amp;lang=en Directorate of Mental Health] in 1998. The Directorate of Mental Health is a part of the [[w:Ministry_of_Health_(Sri_Lanka)|Ministry of Health]] and is responsible for the monitoring and implementation of mental health programs across the country&lt;ref&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?lang=en|title=Home - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;.  As of 2025, the current director of the Directorate of Mental Health is Dr. Chithramalee de Silva&lt;ref name=&quot;:2&quot; /&gt;.

On November 11, 2005, the Mental Health Policy was approved by the Government of Sri Lanka, advocating for establishments of more de-centralized, community-based mental health services across the country. The policy aimed to concisely define the rigorous standards needed to be met for each respected medical professional, including psychiatrists and clinical psychologists&lt;ref&gt;{{Cite journal|last=Rajapakshe|first=Onali Bimalka Wickramaseckara|last2=Mohan|first2=Mohapradeep|last3=Singh|first3=Swaran Preet|date=2023-05|title=Development of adolescent mental health services in Sri Lanka|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10895478/|journal=BJPsych international|volume=20|issue=2|pages=41–43|doi=10.1192/bji.2022.32|issn=2056-4740|pmc=10895478|pmid=38414998}}&lt;/ref&gt;. The policy also included a new position, the &quot;Medical Officer of Mental Health&quot;, tasked with overseeing and assisting in creating community-based mental health services&lt;ref name=&quot;:0&quot; /&gt;. In the same year, the Sri Lankan government began implementing psychological services in state institutions, such as the military&lt;ref name=&quot;:8&quot; /&gt;.  

In 2007, the National Mental Health Advisory Council (NMHAC) was created to serve as an 'advisory' board for the Ministry of Health&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=9&amp;Itemid=220&amp;lang=en|title=Introduction - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;. 

In 2008, the Angoda Mental Hospital was restructured and renamed as the National Institute of Mental Health (NIMH)&lt;ref name=&quot;:7&quot; /&gt;. 

=== Modern-day Sri Lanka ===
[[File:Feeding Children in Sri Lanka.jpg|left|thumb|Despite the noteworthy improvements in mental healthcare services in recent decades, mental health remains a significant issue due to rising poverty. ]]
As of 2025, the Mental Health Act (mental health legislation) has been undergoing development since 2005 and is currently awaiting to be considered for the final stage of approval. This is expected to replace the 1956 Mental Health Ordinance&lt;ref name=&quot;:7&quot; /&gt;.

Currently, there are 7 tertiary care hospitals, 61 adult patient units, 3 child inpatient units, and 1 forensic unit with over 100 psychiatrists all throughout the 22 districts&lt;ref name=&quot;:4&quot; /&gt;. The [[w:Lady_Ridgeway_Hospital_for_Children|Lady Ridgeway Hospital]] in Colombo and the Sirimavo Bandaranayke Specialized Children Hospital in Kandy are specialized in treating children with [[w:Learning_disability|SLD]], [[w:ADHD|ADHD]], [[w:Autism_Spectrum_Disorder|ASD]], and provides family support for patients. As of 2017, 22 rehabilitation centers exist through the country, including 7 alcohol rehab centers&lt;ref name=&quot;:7&quot; /&gt;.   

Despite the impressive advancements in mental healthcare in the last couple of decades, Sri Lanka still suffers significant mental health issues due to increasing poverty levels in the country. The [[w:World_Bank|World Bank]] reported that [https://www.wsws.org/en/articles/2024/04/08/eesc-a08.html the poverty levels in Sri Lanka increased from 11% in 2019 to 26% in 2024], with 60% of Sri Lankan households facing &quot;decreased incomes&quot;&lt;ref&gt;Lakhtakia, Shruti, Atapattu Mudiyanselage, Udahiruni Shashadari Atapat, Walker, Richard Ancrum. ''Sri Lanka Development Update - Bridge to Recovery (English).'' Washington, D.C.: World Bank Group. &lt;nowiki&gt;http://documents.worldbank.org/curated/en/099634104012434919&lt;/nowiki&gt;&lt;/ref&gt;. This was exacerbated by Sri Lanka's excessive foreign debt, economic troubles stemming from [[w:Gotabaya_Rajapaksa|Gotabaya Rajapaksa]]'s presidential term, the COVID-19 pandemic, and the [[w:Russian_invasion_of_Ukraine|ongoing invasion of Ukraine by Russia (2022)]].  

According to [[w:NYU|New York University]] graduate student [https://gc-cuny.academia.edu/NadiaAugustyniak Nadia Augustyniak] in her 2025 overview of Sri Lanka's public mental healthcare system, poverty-induced financial precarity remains a major obstacle to receiving access to mental healthcare services. Even though trauma from adverse weather and conflict is deleterious to mental health, issues originating from every-day struggles, especially struggles related to poverty, could arguably play a more significant role&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.  

== Impact of Conflicts, Terrorism, Political Instability &amp; Natural Disasters ==

=== Sri Lankan Civil War ===
The '''Sri Lankan Civil War''' was a domestic conflict between the Sri Lankan government and the Liberation Tigers of Tamil Eelam (abbreviated as the ''LTTE),'' a militant group formed in the 1970s as a byproduct of rising tensions between the majority Sinhalese and minority Tamil population. The group is considered a terrorist organization&lt;ref&gt;{{Cite web|url=https://www.start.umd.edu/baad/database/liberation-tigers-tamil-eelam-ltte-1998.html|title=BAAD - Liberation Tigers of Tamil Eelam (LTTE) - 1998 {{!}} START.umd.edu|website=www.start.umd.edu|access-date=2025-06-09}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.cfr.org/backgrounder/liberation-tigers-tamil-eelam-aka-tamil-tigers-sri-lanka-separatists|title=Liberation Tigers of Tamil Eelam (aka Tamil Tigers) (Sri Lanka, separatists) {{!}} Council on Foreign Relations|last=Bhattacharji|first=Preeti|website=www.cfr.org|language=en|access-date=2025-06-09}}&lt;/ref&gt;. The LTTE conducted decades of massacres, assassinations of political figures, and suicide bombings to achieve ''[[w:Tamil_Eelam|Tamil Eelam]],'' leading to civilian displacement, infrastructure collapse, and the reduction of mental health services available in the northern region.[[File:DFID-funded, UNHCR emergency shelter tents, in the IDP camp at Menik Farm, Sri Lanka (3694081492).jpg|thumb|350x350px|An IDP camp in Menik Farm, Sri Lanka in 2009 ([https://www.bbc.com/news/world-asia-19703826 now closed]). Suicide rates in IDP camps were three times the general population.]]The civil war mainly affected the northeastern portion of the country, including the [[w:Vanni_(Sri_Lanka)|Vanni region]]. The conflict caused mass destruction to local mental healthcare facilities. Local residents described the conflict as ''varthayal varnicca mudiyathavai'', roughly translating into English as 'beyond description by words'&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|language=en|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;. In 2003, only two psychiatrists were found in the region, operating on extremely limited resources. This furthered long-term trauma and mental health deterioration in the population&lt;ref name=&quot;:5&quot; /&gt;. 

In 2002, the humanitarian organization [https://www.msf.org/ Médecins Sans Frontières] (MSF) conducted an investigation on mental health needs in the [[w:Vavuniya|Vavuniya]] area, the site of intense conflict during the civil war (including the [[w:1985_Vavuniya_massacre|1985 Vavuniya massacre]]), and found that many of the residents suffered from high suicide rates, alcohol abuse, domestic violence, grief, and a &quot;sense of ‘learnt helplessness’&quot;&lt;ref name=&quot;:5&quot; /&gt;. A team from the University of Konstanz in Germany found that 92% of grade school children in the region were exposed to &quot;combat, shelling, and witnessing the death of loved ones&quot;&lt;ref name=&quot;:9&quot; /&gt;. 
[[File:Tractors. Jan 2009 displacement in the Vanni.jpg|left|thumb|350x350px|Displaced civilians evacuating from the Kilinochchi and Mullaitivu Districts due to military campaigns initiated by the Sri Lankan military (January 2009).]]
Additionally, accusations of war crimes have been made against [[w:War_crimes_during_the_final_stages_of_the_Sri_Lankan_civil_war|the Sri Lankan government]]&lt;ref&gt;See also [[w:Sexual violence in the Sri Lankan civil war]].&lt;/ref&gt;. A 2009 HRW report alleged that the Sri Lankan government considered the native Tamil population residing in war zones to be &quot;siding with the LTTE and [therefore, were] treated as combatants&quot;, and that the government conducted numerous shellings of &quot;areas crowded with civilians&quot;&lt;ref&gt;{{Cite journal|date=2009-02-19|title=War on the Displaced|url=https://www.hrw.org/report/2009/02/19/war-displaced/sri-lankan-army-and-ltte-abuses-against-civilians-vanni|journal=Human Rights Watch|language=en}}&lt;/ref&gt;. Furthermore, the LTTE conducted recruitment campaigns on the Vanni population where recruited men, women, and even children with minimal training, were recruited for war efforts.  

Over 200,000 Tamil civilians were moved into [[w:Internally_displaced_persons_in_Sri_Lanka|designated displacement camps during the war]], where conditions were poor&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000}}&lt;/ref&gt;. The suicide rate in these displacement camps was three times the community-level (2002), with a ratio of 103.5 suicides per 10,000 persons, compared to the general population's rate of 37.5 suicides per 10,000 persons. Almost all suicide attempts involved poisonous substances. Other forms of violence included domestic violence and child abuse.  

Local health officials in Vavuniya admitted that mental health concerns were a major problem, but were unable to address these concerns due to a lack of resources and support from the government. During the [[wikipedia:Sri_Lankan_civil_war#2002_peace_process_(2002%E2%80%932006)|brief 2002 ceasefire]], the MSF implemented a &quot;community-based programme&quot; which included &quot;increasing awareness, community strengthening, reinforcing coping-strategies for long-term war-affected communities, and counselling&quot;. The MSF also advocated for restrictions of poisonous substances due its means for suicide attempts, and stressed that &quot;much more [than resettlement]&quot; would need to be done to help alleviate the psychological pain the northern population had faced due to the war&lt;ref&gt;{{Cite journal|last=de Jong|first=Kaz|last2=Mulhern|first2=Maureen|last3=Ford|first3=Nathan|last4=Simpson|first4=Isabel|last5=Swan|first5=Alison|last6=van der Kam|first6=Saskia|date=2002-04|title=Psychological trauma of the civil war in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S0140673602084209|journal=The Lancet|language=en|volume=359|issue=9316|pages=1517–1518|doi=10.1016/S0140-6736(02)08420-9}}&lt;/ref&gt;. The ceasefire ended in 2006 and led to the [[w:Eelam_War_IV|final phase of the civil war]], eventually ending in 2009 with the [[w:https://en.wikipedia.org/wiki/Velupillai_Prabhakaran#Sri_Lankan_Army_Northern_offensive_and_death|death of the LTTE's leader]]. 

'''Post-war'''
[[File:Puttalam district.svg|left|thumb|Puttalam District, unlike its northern counterparts, was largely spared from the intense conflict, possibly explaining the lower rates of common mental disorders (CMDs).]]
The first district-wide cross-sectional multistage cluster sample survey was conducted in the [[w:Jaffna_District|Jaffna District]] shortly after the war ended in 2009. The study's sample included 1517 households and 2 internally displaced peoples camps. With a response rate of 92%, the study found that symptoms for PTSD were found in 7% of participants, symptoms of anxiety were found in 32.6% of participants, and symptoms of depression were found in 22.2% of participants. 2% of respondents were being placed in internally displaced peoples camps at the time of the study, 29.5% were freshly resettled from the internally displaced peoples camps, and the rest of the participants (68.5%) were never placed into camps. In comparison to residents who were never placed into camps, participants that were actively held in camps generally reported more symptoms of PTSD, anxiety, and depression. The researchers also found that women were especially vulnerable to deteriorating mental health conditions. This was explained by two factors: women having to assume the roles of both the father and the mother in the family setting after the, either voluntary or forced, departure of their husband to war, and sexist violence&lt;ref&gt;{{Cite journal|last=Husain|first=Farah|last2=Anderson|first2=Mark|last3=Lopes Cardozo|first3=Barbara|last4=Becknell|first4=Kristin|last5=Blanton|first5=Curtis|last6=Araki|first6=Diane|last7=Kottegoda Vithana|first7=Eeshara|date=2011-08-03|title=Prevalence of War-Related Mental Health Conditions and Association With Displacement Status in Postwar Jaffna District, Sri Lanka|url=https://doi.org/10.1001/jama.2011.1052|journal=JAMA|volume=306|issue=5|pages=522–531|doi=10.1001/jama.2011.1052|issn=0098-7484}}&lt;/ref&gt;. A 2013 study on adult patients in [https://www.ncbi.nlm.nih.gov/books/NBK232631/ primary care settings] (divisional hospitals, primary medical care units) found major depression to be significantly higher in females (5.1%) than males (3.6%), bolstering the findings from the 2009 study&lt;ref&gt;{{Cite journal|last=Senarath|first=Upul|last2=Wickramage|first2=Kolitha|last3=Peiris|first3=Sharika Lasanthi|date=2014-03-24|title=Prevalence of depression and its associated factors among patients attending primary care settings in the post-conflict Northern Province in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/1471-244X-14-85|journal=BMC Psychiatry|language=en|volume=14|issue=1|pages=85|doi=10.1186/1471-244X-14-85|issn=1471-244X|pmc=3987835|pmid=24661436}}&lt;/ref&gt;.

Muslims in Northern Sri Lanka also faced violence and discrimination during the conflict. Most notable incidents include [[w:Expulsion_of_Muslims_from_the_Northern_Province_of_Sri_Lanka|the October 1990 expulsion of Muslims from the North to the Puttalam District or Jaffna]] and the [[w:Kattankudy_mosque_massacre|1990 Kattankudy mosque massacre]]. The only study testing the displaced Muslim population post-civil war was completed in 2011, where a cross-sectional survey of 450 internally displaced people or people born into displacement (ages 18 - 65) revealed 18.8% of the sample suffering from common mental health disorders (CMD), including [[w:Somatoform_disorder|somatoform disorder]] (14%), &quot;other depressive syndromes&quot; (7.3%), major depression (5.1%), and anxiety disorder (2.8%). The percentages found in this study for somatoform disorder and major depression were &quot;considerably higher&quot; than the national percentages, though the researchers noted that the prevalence of CMD was lower in comparison to other countries marred with conflict, including Palestine (40.3%) and Ethiopia (27.8%). The researchers explained that the lower rate of CMD may be attributed to the [[w:Puttalam_District|serenity of the post-settlement destination]], as conflict was mainly centered in the North and East. In contrast to earlier findings, this study did not observe a higher prevalence of CMDs among women, although increased rates of somatoform disorders were noted (though the researchers did not reveal the data behind this)&lt;ref&gt;{{Cite journal|last=Siriwardhana|first=Chesmal|last2=Adikari|first2=Anushka|last3=Pannala|first3=Gayani|last4=Siribaddana|first4=Sisira|last5=Abas|first5=Melanie|last6=Sumathipala|first6=Athula|last7=Stewart|first7=Robert|date=2013-05-22|title=Prolonged Internal Displacement and Common Mental Disorders in Sri Lanka: The COMRAID Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0064742|journal=PLOS ONE|language=en|volume=8|issue=5|pages=e64742|doi=10.1371/journal.pone.0064742|issn=1932-6203|pmc=3661540|pmid=23717656}}&lt;/ref&gt;.

Research on the mental state of combatants has been limited, but a post-war 2009 study done between soldiers of the [[w:Sri_Lanka_Army_Special_Forces_Regiment|Special Forces]] and regular soldiers showed higher levels of exposure to traumatic events for units of the Special Forces, yet the former exhibited significantly less symptoms of CMDs compared to the latter. The authors of this study, [https://scholar.google.co.uk/citations?user=cVKEBdwAAAAJ&amp;hl=en&amp;oi=ao Raveen Hanwella] and [https://scholar.google.co.uk/citations?user=ZRj74qMAAAAJ&amp;hl=en&amp;oi=sra Varuni de Silva], offered the camaraderie of the military unit as an explanation for the discrepancy&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|date=2012-08|title=Mental health of Special Forces personnel deployed in battle|url=https://pubmed.ncbi.nlm.nih.gov/22038567|journal=Social Psychiatry and Psychiatric Epidemiology|volume=47|issue=8|pages=1343–1351|doi=10.1007/s00127-011-0442-0|issn=1433-9285|pmid=22038567}}&lt;/ref&gt;. A follow-up study was completed by the pair (with the addition of former Director-General of the Health Services of the Sri Lanka Navy [[w:Nicholas_Jayasekera|Nicholas Jayasekera]]), where the findings were similar, though the statistically significant bridge between the two cohorts in the previous study evaporated in the follow-up study. This may be due to the significant decline in mental health problems observed in the regular unit forces, potentially reflecting resilience in the aftermath of the conflict&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=Jayasekera|first2=Nicholas E. L. W.|last3=Silva|first3=Varuni A. de|date=2014-09-25|title=Mental Health Status of Sri Lanka Navy Personnel Three Years after End of Combat Operations: A Follow Up Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0108113|journal=PLOS ONE|language=en|volume=9|issue=9|pages=e108113|doi=10.1371/journal.pone.0108113|issn=1932-6203|pmc=4177866|pmid=25254557}}&lt;/ref&gt;. Amputees or soldiers with spinal injuries exhibited drastically different numbers, with approximately 40% of nearly 100 male-veterans in a post-war 2009 study displaying PTSD-like symptoms&lt;ref&gt;{{Cite journal|last=Abeyasinghe|first=N. L.|last2=de Zoysa|first2=P.|last3=Bandara|first3=K.M.K.C.|last4=Bartholameuz|first4=N. A.|last5=Bandara|first5=J. M.U.J.|date=2012-05-01|title=The prevalence of symptoms of Post-Traumatic Stress Disorder among soldiers with amputation of a limb or spinal injury: A report from a rehabilitation centre in Sri Lanka|url=https://doi.org/10.1080/13548506.2011.608805|journal=Psychology, Health &amp; Medicine|volume=17|issue=3|pages=376–381|doi=10.1080/13548506.2011.608805|issn=1354-8506|pmid=21942815}}&lt;/ref&gt;.

About a decade after the conflict ceased, a few notable studies have emerged to help guide understanding on the longer-term mental health effects on victims of the civil war. 

From July 2019 to October 2020, a study conducted on 585 local adolescents (ages 12-19) in the Vavuniya district revealed that despite 15.6% of the statistic having faced one or more war-related events, only 3.9% of the participants had moderate to severe depression. In addition to considerably low depression rates, only 5.7% of participants age 17+ were found to have moderate to severe hopelessness&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000|pmc=10472617|pmid=37653394}}&lt;/ref&gt;. The authors referenced a 2010 observation by psychiatrist [https://us.sagepub.com/en-us/nam/author/daya-somasundaram Daya Somasundaram], who noted that many Tamil IDPs presented &quot;remarkable resilience and post-traumatic growth&quot; after the civil war—an outcome he attributed to the close-knit, family-centered nature of Tamil communities&lt;ref&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;.  

However, findings originating from a 2019 study, undertook by several faculty members from the University of Kelaniya, the University of Jaffna, the [[w:Gampaha_Wickramarachchi_University_of_Indigenous_Medicine|Gampaha Wickramarachchi University of Indigenous Medicine]], and the [https://onur.gov.lk/ Office for National Unity and Reconciliation (ONUR)] in Jaffna, found contrasting results. Out of 336 participants from districts which faced significant ramifications of the conflict (Jaffna, Kilinochchi, Mullaithivu, Vavuniya, and Mannar districts), 50.5% had extreme anxiety symptoms and 36.5% exhibited &quot;extremely severe&quot; symptoms of depression. 92.5% of families in the sample experienced suicidal ideation, with an observed negative correlation between trauma exposure and life satisfaction with families. Drug abuse (86.2%) and alcohol abuse (84.5%) were the two highest problematic behaviors recorded on a community-level, suggesting that the negative consequences of the civil war still persist, possibly on a substantial scale than previously recognized, in Tamil communities residing in the North&lt;ref&gt;{{Cite journal|last=Thamotharampillai|first=Umaharan|last2=Perera|first2=Ruwanthi|last3=Wickremasinghe|first3=Rajitha|last4=Williams|first4=Shehan|last5=Vijayasangar|first5=Thedsanamoorthy|last6=Sivatharsan|first6=Balasubramaniam|last7=Hilbert|first7=Vanceline|last8=Somasundaram|first8=Daya|date=2025-05-06|title=Collective Trauma- Psychosocial consequences of war in northern Sri Lanka 10 years on, a mixed methods study|url=https://www.sciencedirect.com/science/article/pii/S2666560325000696|journal=SSM - Mental Health|pages=100457|doi=10.1016/j.ssmmh.2025.100457|issn=2666-5603}}&lt;/ref&gt;. Further research should be conducted on Northern Tamil populations to assess the extent of mental health issues stemming from the conflict. 

In 2019, [https://www.researchgate.net/scientific-contributions/R-M-M-Monaragala-2087692299 Dr. R. M. M. Monaragala] conducted a study on 1,845 soldiers with combat experience, finding that 3.9% of the sample suffered from PTSD. Dr. Monaragala noted that &quot;probable depression, fatigue, aggression, and family history of mental disorder&quot; were correlative of PTSD presence. He suggested that &quot;screening and psychosocial intervention[s]&quot; could alleviate CMDs of former combatants&lt;ref&gt;{{Cite journal|last=Monaragala|first=R. M. M.|date=2024-04-19|title=Exploring the effects of the past civil war in terms of the prevalence and associating factors of PTSD|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v14i2.8465|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=14|issue=2|doi=10.4038/sljpsyc.v14i2.8465|issn=2012-6883}}&lt;/ref&gt;. 

=== 2004 Boxing Day Tsunami ===
The '''2004 Boxing Day Tsunami''' was a natural disaster where a tsunami spawned off a 9.2–9.3 magnitude earthquake off the coast of Aceh in Indonesia on December 26. The tsunami greatly affected the coastlines of the country, with the death toll reaching to around 35,000 deaths. In addition, 90,000 houses were destroyed and 516,000 people were forced to migrate due to severe infrastructural damage&lt;ref name=&quot;:5&quot; /&gt;. It stands as the [http://www.china.org.cn/english/features/tsunami_relief/119821.htm worst natural disaster to have ever hit Sri Lanka].  
[[File:Tsunami relief 2004 02.jpg|thumb|300x300px|Volunteers from [[w:Royal_College,_Colombo|Royal College in Colombo]] assisting in tsunami relief efforts (Sarvodaya Headquarters, Moratuwa).]]
A survey conducted on schoolchildren (ages 8-14) in Manadkadu (a Tamil-majority village in the northern coast), [[w:Kosgoda|Kosgoda]] (western coast), and [[w:Galle|Galle]] (southern coast), just a few weeks after the tsunami hit Sri Lanka, revealed that 33.8%, 13.9%, and 38.8% of children interviewed exhibited signs of PTSD (according to the DSM-IV's criteria), respectively (minus the time criteria, as the DSM-IV does not permit diagnosis of PTSD within 4 weeks of a traumatic incident). The loss of family members and exposure to previously traumatic incidents appeared to be highly correlate with PTSD development&lt;ref&gt;{{Cite journal|last=Neuner|first=Frank|last2=Schauer|first2=Elisabeth|last3=Catani|first3=Claudia|last4=Ruf|first4=Martina|last5=Elbert|first5=Thomas|date=2006|title=Post-tsunami stress: A study of posttraumatic stress disorder in children living in three severely affected regions in Sri Lanka|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/jts.20121|journal=Journal of Traumatic Stress|language=en|volume=19|issue=3|pages=339–347|doi=10.1002/jts.20121|issn=1573-6598}}&lt;/ref&gt;. 

Many victims in the Jaffna area suffered with &quot;[https://www.psychiatry.org/patients-families/prolonged-grief-disorder pathological grief], phobias, depression and PTSD&quot; post-tsunami. Schizophrenia in the Jaffna Tamil community, which had already suffered elevated prevalence of PTSD prior to the tsunami, had worsened—highlighting the need for specialized care in response to cumulative exposures to chronic and acute traumas. In a study published in ''International Psychiatry'' (2006), Jaffna-based researchers noted that, contrary to their initial inclinations, there was not a &quot;large[r] (than expected) rise in [the] number of people&quot; seeking mental health support 3 months after the tsunami. However, 10 months after the disaster, the researchers anticipated that &quot;more psychiatric disorders&quot; would emerge due to &quot;very little rebuilding [efforts]&quot; and an apparent &quot;unfairness in the aid system&quot;.&lt;ref&gt;{{Cite journal|last=Somasundaram|first=D. J.|last2=Yoganathan|first2=S.|last3=Ganesvaran|first3=T.|date=1993-09|title=Schizophrenia in northern Sri Lanka|url=https://pubmed.ncbi.nlm.nih.gov/7828234|journal=The Ceylon Medical Journal..|volume=38|issue=3|pages=131–135|issn=0009-0875|pmid=7828234}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Danvers|first=K.|last2=Sivayokan|first2=S.|last3=Somasundaram|first3=D. J.|last4=Sivashankar|first4=R.|date=2006-07|title=Ten months on: qualitative assessment of psychosocial issues in northern Sri Lanka following the tsunami|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6734678/|journal=International Psychiatry: Bulletin of the Board of International Affairs of the Royal College of Psychiatrists|volume=3|issue=3|pages=5–8|issn=1749-3676|pmc=6734678|pmid=31507850}}&lt;/ref&gt; 

At the February 2005 ''After the Tsunami: Mental Health Challenges to the Community for Today and Tomorrow'' conference in Thailand, [https://www.researchgate.net/profile/Chandanie-Hewage Dr. Chandanie Hewage] of the [[w:University_of_Ruhuna|University of Ruhuna]] commentated that measures taken to assist the affected were &quot;not coordinated&quot; due to poor &quot;communication systems and road [conditions].&quot; Regardless, efforts were continued by the government and health professionals to alleviate the struggles the victims were facing, including the psychological ramifications of the disaster. 

Several issues in the delivery of these services were highlighted by Dr. Hewage, including poor maintenance of health records, lack of awareness on drug consumption by the patients themselves, and shortages of health professionals. Dr. Hewage points out that personnel had &quot;little&quot; mental health training prior to the disaster, suggesting increased &quot;research&quot; and adequate &quot;provision[ing] and training of staff&quot; for the long-term&lt;ref&gt;{{Cite journal|last=Davidson|first=Jonathan R. T.|date=2006|title=Foreword. After the tsunami: mental health challenges to the community for today and tomorrow|url=https://pubmed.ncbi.nlm.nih.gov/16602809|journal=The Journal of Clinical Psychiatry|volume=67 Suppl 2|pages=3–8|issn=0160-6689|pmid=16602809}}&lt;/ref&gt;. With inadequate documentation, no systematic procedures in place, and insufficient personnel, tsunami victims with mental health concerns may not receive the services they need, further compacting neuropsychological ailments.  

In 2008 (about 3-4 years after the tsunami), researchers in the hard-hit village of [[w:Peraliya|Peraliya]] (Galle District) found that from a sample of approximately 90 adults, 25% suffered from moderate–severe PTSD, with women scoring &quot;above the cut-off for anxiety&quot; and reporting more &quot;somatic symptoms&quot;, though researchers inferred that the PTSD rate found in the study may be influenced by other factors, including war or economic hardship&lt;ref&gt;{{Cite journal|last=Hollifield|first=Michael|last2=Hewage|first2=Chandanie|last3=Gunawardena|first3=Charlotte N.|last4=Kodituwakku|first4=Piyadasa|last5=Bopagoda|first5=Kalum|last6=Weerarathnege|first6=Krishantha|last7=Group|first7=International Post-Tsunami Study|date=2008-01|title=Symptoms and coping in Sri Lanka 20–21 months after the 2004 tsunami|url=https://www.cambridge.org/core/journals/the-british-journal-of-psychiatry/article/symptoms-and-coping-in-sri-lanka-2021-months-after-the-2004-tsunami/CB33752239AF362A0BFD55B3668D60B0|journal=The British Journal of Psychiatry|language=en|volume=192|issue=1|pages=39–44|doi=10.1192/bjp.bp.107.038422|issn=0007-1250}}&lt;/ref&gt;.  

=== 2019 Easter Bombings ===
The '''2019 Easter Bombings''' were a series of coordinated attacks perpetrated by the Islamic extremist group, [[w:National_Thowheeth_Jama'ath|National Thowheeth Jama'ath]], on April 21, 2019. The attack targeted three churches and three hotels in the Colombo area, killing nearly 300 people and injuring over 500. The attacks were also attributed to the incompetency of the Sri Lankan government, who ignored [https://www.bbc.com/news/world-asia-48044636 multiple warnings preceding the attacks]. The attacks negatively affected the Sri Lankan Catholic community and further weakened relations between the major religious groups&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;.  

In the aftermath of the attacks, professionals in the [[w:Gampaha_District|Gampaha District]] resorted to &quot;low-cost methodologies&quot; for children and adolescents affected by the attack, as a &quot;severe shortage&quot; of children and adolescent mental health experts were exposed&lt;ref&gt;{{Cite journal|last=Chandradasa|first=Miyuru|last2=Rathnayake|first2=Layani C|last3=Rowel|first3=Madushi|last4=Fernando|first4=Lalin|date=2020-06-01|title=Early phase child and adolescent psychiatry response after mass trauma: Lessons learned from the Easter Sunday attack in Sri Lanka|url=https://doi.org/10.1177/0020764020913314|journal=International Journal of Social Psychiatry|language=EN|volume=66|issue=4|pages=331–334|doi=10.1177/0020764020913314|issn=0020-7640}}&lt;/ref&gt;. In a qualitative study of 8 survivors of the attacks receiving grief counseling, [[w:University_of_Ruhuna|University of Ruhuna]] assistant professor [https://www.researchgate.net/profile/Virasha-Godakanda Virasha Godakanda] observed that 70% of the sample size expressed a lack of confidence in adequate mental health interventions from the government, reducing the quality of such services. Professor Godakanda strongly endorsed for &quot;culturally-sensitive&quot; programs, a diversity in therapeutic approaches (including nature-based therapy), and &quot;prolonged investigations&quot; to track developments in mental health resources and impacts of implemented interventions&lt;ref&gt;{{Cite journal|last=Godakanda|first=Virasha|date=2025-01-29|title=A GRIEF COUNSELING INTERVENTION AFTER THE MASS TRAUMA: LESSONS LEARNED FROM THE VICTIMS OF THE EASTER SUNDAY ATTACK IN SRI LANKA|url=https://kjmr.com.pk/kjmr/article/view/216|journal=Kashf Journal of Multidisciplinary Research|language=en|volume=2|issue=01|pages=13–32|doi=10.71146/kjmr216|issn=3007-200X}}&lt;/ref&gt;. 

A few weeks following the attacks, Muslims in Sri Lanka were subjected to [[w:2019_anti-Muslim_riots_in_Sri_Lanka|violent, coordinated riots]] masterminded by Sinhalese national forces&lt;ref&gt;{{Cite journal|last=Mujahidin|first=Muhammad Saekul|date=2023-07-03|title=Extremism and Islamophobia Against the Muslim Minority in Sri Lanka|url=https://www.ajis.org/|journal=American Journal of Islam and Society|language=en|volume=40|issue=1-2|pages=213–241|doi=10.35632/ajis.v40i1-2.3135|issn=2690-3741}}&lt;/ref&gt;. Riots were mainly centered in the [[w:Kurunegala_District|Kurunegala]], Gampaha, and [[w:Kandy_District|Kandy]] Districts. At least [https://www.aljazeera.com/news/2019/5/21/in-sri-lanka-muslims-say-sinhala-neighbours-turned-against-them one confirmed death was reported]. Calls for vague ''niqab'' and ''burqa'' bans were increasingly prominent, eventually leading to the 2021 burqa ban by the Sri Lankan government. Pakistani and Afghani refugees fleeing religious persecution in Negombo were forced to be &quot;made refugees again&quot; after local protests were orchestrated against their settlement. Anti-Muslim sentiment was &quot;unleashed online, in the law, and on the street&quot;&lt;ref&gt;{{Cite book|title=CARTOGRAPHIC JOURNEY OF RACE, GENDER AND POWER: global identity|date=2021|publisher=CAMBRIDGE SCHOLARS PUBLIS|isbn=978-1-5275-6965-2|location=S.l.}}&lt;/ref&gt;. Albeit its relevancy to the attacks, no in-depth mental health studies have took place on the minority Muslim population following the Easter bombings. Further research is imperative in exploring the sustained psychological effects of Islamophobia and its effect on the Muslim minority community in the aftermath of the 2019 Easter attacks. 

Literature on the impact of the 2019 Easter Bombings on mental health is limited and further research should be conducted.  

=== 2019-2024 Economic Crisis ===
The '''2019-2024 Economic Crisis''' refers to a 5 year period where the Sri Lankan economy experienced significant inflation and an abrupt hike in prices on basic, everyday items. It is the worse economic crisis the country has faced since the Sri Lankans were granted independence in 1948. Schools in Sri Lanka were forced to postpone examinations due to paper shortages. Gas shortages led to long lines at gas stations, some lasting for days, throughout the island. Shortages in electricity, cooking gas, and aviation feul were additional consequences of the economic crisis.

Healthcare workers faced a barrage of impediments in their line of work during the crisis, including a lopsided work-life balance due to unprecedented demand, increased stress and mental fatigue from a lack of resources and personnel, unhealthy coping mechanisms, job dissatisfaction, and a reduction in work quality. Such effects perpetuated a self-enforcing cycle of psychologically distressed mental healthcare workers providing subpar services, affecting patients and amplifying mental health issues experienced by both the workforce and their patients&lt;ref&gt;{{Cite journal|last=Dilogini|first=S.|last2=Grace|first2=H. H.|last3=Thasika|first3=T.|date=2024|title=Exploring The Mental Health and Well-Being of Public Healthcare Workers (HCWs) Amid Economic Crisis in Sri Lanka|url=http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/11092|language=en|publisher=Chartered Institute of Personnel Management}}&lt;/ref&gt;. 

Medical students from the Faculty of Medicine at the University of Colombo reported that the economic crisis forced abrupt changes in dietary consumption, increased hopelessness in the future, increased stress and anxiety, and a decrease in interest in pursuing a &quot;clinical post-graduate career&quot;&lt;ref&gt;{{Cite journal|last=Adikaranayake|first=Pesala Randika|last2=Perera|first2=Anusha Nimrod|last3=Nilaweera|first3=Akhila Imantha|last4=Fernando|first4=Desha Rajni|last5=Wijayaratne|first5=Dilushi Rowena|date=2025-07-01|title=Effects of Sri Lankan economic crisis on health, lifestyle and education of medical students in Faculty of Medicine, University of Colombo – an online survey|url=https://doi.org/10.1186/s12909-025-07506-y|journal=BMC Medical Education|language=en|volume=25|issue=1|pages=938|doi=10.1186/s12909-025-07506-y|issn=1472-6920|pmc=12211748}}&lt;/ref&gt;. 283 government-school teachers completed a web-based cross-sectional survey in April 2024, with majority of the participants reporting a severe reduction in monthly income &amp; 1/3 of participants exhibiting &quot;clinical levels of psychological distress&quot;&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C. P.|last2=Senarathne|first2=D. L. P.|last3=Fernando|first3=M. S.|last4=Senevirathne|first4=S. P.|date=2025-05-28|title=Examining the economic burden and mental health distress among government school teachers in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/s40359-025-02921-8|journal=BMC Psychology|language=en|volume=13|issue=1|pages=572|doi=10.1186/s40359-025-02921-8|issn=2050-7283}}&lt;/ref&gt;. A study published in that same year reported that out of 261 nurses working in teaching hospitals, 91.6% were forced to allocate their finances to strictly &quot;general needs&quot;, while more than 50% looked into international opportunities for employment. Notably, the study reported an overall near &quot;twofold greater&quot; rate of depression, anxiety, and stress compared to previous studies on nurses in Sri Lanka&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C.P|last2=Senarathne|first2=L.|last3=Fernando|first3=M.|date=2024-04-01|title=Exploring the Association Between Behavioural Modification in Response to the Prevailing Economic Crisis and Mental Health Outcomes of Nurses from Teaching Hospitals, Sri Lanka|url=https://doi.org/10.1177/23779608241272679|journal=SAGE Open Nursing|language=EN|volume=10|pages=23779608241272679|doi=10.1177/23779608241272679|issn=2377-9608|pmc=11311183}}&lt;/ref&gt;. 

The detrimental effects the crisis has had on the mental health sector reveal a concerning area of underappreciation and under compensation towards a critical sector for the well-being of the country. Adequate staffing, increased funding, and an improved work-life balance should be emphasized for the workers of health sector of the country.  

== Present-Day Challenges ==

=== Ethnic tension ===
Despite the ending of the Sri Lankan civil war and the introduction of pluralist policies (such as the [https://srilankaembassy.fr/sites/default/files/files/media/pdf/NationalPolicy-English.pdf 2017 National Policy on Reconciliation and Coexistence] under the Sirisena administration), tensions amongst members of the ethnic groups still persist. Evidence of these tensions was found in a 2022 study conducted in the Ratnapura district, where religious leaders expressed skepticism through semi-structured interviews on &quot;conflict transformation&quot;. A Tamil citizen of the Ratnapura community recounted that they were forced to &quot;hide in jungles&quot; and consume &quot;dirty water in drainage[s]&quot; due to scarcity of food and drinkable water as a result of the conflict. In certain personal accounts, ethnic conflicts appear to affect the social behavior and identity of the majority ethnic group. One Sinhala participant recounted his objection to the war-time retaliatory destruction of a shop run by a Tamil shopkeeper was met with interrogative questions about &quot;whether [he was] Sinhalese or not&quot;. Both accounts convey interethnic tensions stemming from decade-long conflicts&lt;ref&gt;Jayathilaka, Aruna &amp; Gamage, Sayuri. (2024). Role of Buddhist and Hindu Religious Leaders Role of Buddhist and Hindu Religious Leaders in the Post-War Conflict Transformation Process: A Study Based on Rathnapura District in Srilanka. ''Retrieved from'' https://gandhimargjournal.org/wp-content/uploads/2024/09/Volume-46-Issue-1-April-June-2024.pdf#page=66&lt;/ref&gt;. 

Beyond individual accounts and the official end of the civil war, the minority groups in the country continue to feel ostracized. The Sri Lankan Tamil population remains dissatisfied with the Sri Lankan government due to their alleged lack of accountability of perpetrators of war crimes and lack of information on the whereabouts of [[w:Enforced_disappearances_in_Sri_Lanka|thousands of enforced disappearances]] that took place from the 1980s. Additionally, rising anti-Muslim sentiment in recent years has contributed to increased ethnic tensions, a stark contrast to the previous centuries of peaceful co-existence between the groups. 
[[File:Bodu Bala Sena symbol.svg|thumb|The symbol for Bodu Bala Sena, a nationalistic Sinhala Buddhist group criticized for catalyzing ethnic tensions in Sri Lanka.]]
Laws passed by the Sri Lankan government, such as the [[w:Prevention_of_Terrorism_Act_(Sri_Lanka)|Prevention of Terrorism Act]] and [[wikipedia:Anti-conversion_law#Sri_Lanka|anti-conversion laws]], have forced the United States Commission on International Religious Freedom to label Sri Lanka as a nation that &quot;[engages] or [tolerates] severe violations of religious freedom&quot; in their 2024 report. The government has been criticized by human rights organizations for &quot;disproportionately targeting religious minorities&quot;&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;. Additionally, the implementation of the three dominant languages, English, Sinhala, and Tamil, across formal education and government services have been lackadaisical, narrowing opportunities of foundational social interactions between the groups. Persistent discrimination and prejudice towards minority groups can lead to an array of complex and self-deprecating mental health issues.   

Efforts to mitigate ethnic tensions include strategies like [[w:Community-based_participatory_research|community-based participatory research]] (CBPR), task-sharing, and securing online mental health services in order to expand mental health services. However, the implementation of evidence-based plans has been met with difficulty due to inaccessibility, high costs, and shortages of adequately-trained personnel.  

Movements aiming for improved intra group and inter group coexistences, such as the Jaffna People’s Forum for Coexistence, should be emphasized on a systematic and multi-level basis, including but not limited to education, public sectors, and within communities. Pluralistic values are encouraged to be emphasized across both private and public schools to foster cultural sensitivity and tolerance. Measures should be taken against groups criticized for promoting sectarian hostility, such as the [[w:Bodu_Bala_Sena|Bodu Bala Sena]]. 

=== Poverty ===
It has been proven that poverty significantly increases the chances of developing mental illnesses. This is further amplified by possible discrimination&lt;ref&gt;{{Cite journal|last=Knifton|first=Lee|last2=Inglis|first2=Greig|date=2020-10|title=Poverty and mental health: policy, practice and research implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7525587/|journal=BJPsych bulletin|volume=44|issue=5|pages=193–196|doi=10.1192/bjb.2020.78|issn=2056-4694|pmc=7525587|pmid=32744210}}&lt;/ref&gt;. Poverty also affects the ability for individuals with mental health concerns to receive the treatment they need. Due to the repercussions of the economic crisis, clients in Sri Lanka could not attend further counseling sessions&lt;ref name=&quot;:8&quot; /&gt;. Poverty from 2021 to 2022 [https://databankfiles.worldbank.org/public/ddpext_download/poverty/987B9C90-CB9F-4D93-AE8C-750588BF00QA/current/Global_POVEQ_LKA.pdf reportedly doubled], with future forecasts predicting the poverty line to &quot;remain above 25 percent&quot;. Suicide has been empirically linked to economic hardships in previous studies&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. A 2013 study done on suicidal patients in [[w:Batticaloa_Teaching_Hospital|Batticaloa Teaching Hospital]] revealed 76% of patients who attempted suicide were from rural areas while 15% were from urban areas&lt;ref&gt;{{Cite book|url=http://ir.lib.seu.ac.lk/handle/123456789/1457|title=The influence of common risk factors for the patient with attempted suicide hospitalized at the teaching hospital, Batticaloa|last=Kisokanth|first=G.|last2=Najeem|first2=M. M.|last3=Karunakaran|first3=K. E.|date=2014-08-02|publisher=South Eastern University of Sri Lanka, University Park, Oluvil #32360, Sri Lanka|isbn=978-955-627-053-2|language=en-US}}&lt;/ref&gt;. The Sri Lankan government should consider the economical impacts that poverty has on mental health and implement ways to aid poverty-stricken individuals with mental health concerns.   

=== Stigmas ===
Stigma consists of the &quot;combined effect of prejudice, ignorance and discrimination.&quot;&lt;ref name=&quot;:10&quot;&gt;{{Cite web|url=http://www.researchgate.net/publication/233990797_The_Stigma_of_Mental_Illness_in_Sri_Lanka_The_Perspectives_of_Community_Mental_Health_Workers|title=(PDF) The Stigma of Mental Illness in Sri Lanka: The Perspectives of Community Mental Health Workers|website=ResearchGate|language=en|access-date=2025-07-25}}&lt;/ref&gt;.

A 2012 interview consisting of nine participants (two doctors, three nurses, one occupational therapist, one development worker, and two volunteers) revealed a number of concerning societal viewpoints on individuals with mental health concerns. The interviews revealed that negative judgements were not only levied against the individual with the mental illness, but also the family. Families hid mentally ill family members from the public to avoid &quot;shame&quot; and possible hinderances in marriage proposals. Views that mentally ill individuals were &quot;violent&quot; served as the motivating factor behind socially isolating those with mental illness from their communities. Interviewees mentioned that individuals dealing with mental health challenges would be attacked with stones and called &quot;derogatory names.&quot; A lack of community awareness regarding mental health and negative portrayals of mentally ill individuals in media exacerbates stigmatization, though the researchers commented that the media was &quot;improving&quot; in their depiction of mental illness. Beliefs that illnesses are caused by &quot;spirits&quot; can be problematic for individuals dealing with mental health issues and suggests poor mental health awareness. Mental health workers themselves believed that they were being stigmatized, as mental health was reportedly not taken as seriously as physical health. Despite the intriguing perspectives provided, the small sample size and usage of snow sampling raise questionable concerns regarding the generalizability of the results&lt;ref name=&quot;:10&quot; /&gt;. 

Improving media portrayal of subjects concerning mental health and involving community members in interventions dealing with mental health issues are ways that could destigmatize mental health amongst communities in Sri Lanka. Tying collaborations between allopathic services and traditional healers instead of having these two services work individually could enhance engagement between traditional medicine and Western medicine.

=== Suicide Trends &amp; Risk Factors ===
Suicide is defined as &quot;the act of killing oneself deliberately, initiated and performed by the person concerned in the full knowledge or expectation of its fatal outcome&quot;&lt;ref name=&quot;:11&quot;&gt;{{Cite book|title=The neuroscience of suicidal behavior|last=Heeringen|first=Kees van|date=2018|publisher=Cambridge University Press|isbn=978-1-316-60290-4|series=Cambridge fundamentals of neuroscience in psychology|location=Cambridge, United Kingdom New York, NY, USA Port Melbourne, VIC, Australia New Delhi, India Singapore}}&lt;/ref&gt;. Although Sri Lanka has seen a significant reduction in suicide rates from the mid 1990s, largely stemming from its ban on extremely toxic pesticide products, suicide and self harm remains a significant issue. The suicide rate per 100,000 people increased from 14.0 in 2019 to [https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide 15.0 in 2022] (according to WHO). On average, 27 males per 100,000 males and 5 females per 100,000 females committed suicide in 2022&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. Hanging appears to be the most used method for suicide for both males and females, with studies revealing a steady increase in recent years&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Bandara|first=Piumee|last2=Wickrama|first2=Prabath|last3=Sivayokan|first3=Sambasivamoorthy|last4=Knipe|first4=Duleeka|last5=Rajapakse|first5=Thilini|date=2024-04-17|title=Reflections on the trends of suicide in Sri Lanka, 1997–2022: The need for continued vigilance|url=https://journals.plos.org/globalpublichealth/article?id=10.1371/journal.pgph.0003054|journal=PLOS Global Public Health|language=en|volume=4|issue=4|pages=e0003054|doi=10.1371/journal.pgph.0003054|issn=2767-3375|pmc=11023397|pmid=38630779}}&lt;/ref&gt;. 

From 2023 to 2024, a group of researchers from the [[w:Eastern_University,_Sri_Lanka|Eastern University in Sri Lanka]] assessed 828 patients admitted to the Teaching Hospital in [[w:Batticaloa,_Sri_Lanka|Batticaloa, Sri Lanka]] for attempted suicide. They concluded that suicide prevention programs should be attuned to younger people (ages 15 to 35 in the study), emphasize the importance of education and reducing unemployment, and increase social support in the Tamil community. Despite accounting for other factors that could lead to suicidal ideation (ie, poverty), the results from this study suffer in external validity as 90% of the patients were Tamil and over 50% were between 16 and 25 years. In addition, correlations between suicide and unemployment rates have been questioned, with [[w:Austerity|austerity]] being a more reliable indicator of suicide rates than unemployment rates&lt;ref name=&quot;:11&quot; /&gt;. Further comprehensive studies on risk factors relating to suicide should be studied to examine correlations between unemployment rates and austerity measures.  

The WHO suggests implementing evidence-based suicide prevention programs, such as [https://www.who.int/initiatives/live-life-initiative-for-suicide-prevention LIVE LIFE], to reduce the national suicide rate&lt;ref&gt;{{Cite web|url=https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide|title=World Suicide Prevention day 2024 “Changing the Narrative on Suicide”|website=www.who.int|language=en|access-date=2025-07-29}}&lt;/ref&gt;. Media depictions of suicidal methods, such as hanging, can lead to sensationalism and the media should be cautious of such displays in movies and TV shows&lt;ref name=&quot;:12&quot; /&gt;. Awareness of depression and other mental health issues can serve as a safeguard against suicidal ideation in Sri Lankan men and women.  

== Role of Religion ==
According to the last demographic report (2012), 70.2% of Sri Lankans are Buddhist, 12.6% are Hindus, 9.7% are Muslims, and 7.4% are Christians. The Theravada Buddhist community makes up the majority in several provinces throughout the country&lt;ref&gt;{{Cite web|url=https://www.state.gov/reports/2022-report-on-international-religious-freedom/sri-lanka/|title=Sri Lanka|website=United States Department of State|language=en-US|access-date=2025-08-07}}&lt;/ref&gt;. Religion, especially Theravada Buddhism, has had a significant influence on not only the historical treatment of mental health in the country, but also everyday life&lt;ref name=&quot;:15&quot; /&gt;. The [[w:Mahāvaṃsa|''Mahāvaṃsa'']] details hospitals treating patients suffering from mental health issues as early as the 4th century BC. Additionally, the 1700s Nayaka king [[w:Kirti_Sri_Rajasinha|Kirthi Sri Rajasinghe]] detailed the implementation of Buddhist philosophy in psychiatry&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. De|date=2017-12-05|title=Development of civil commitment statutes (laws of involuntary detention and treatment) in Sri Lanka: a historical review|url=https://mljsl.sljol.info/articles/10.4038/mljsl.v5i1.7351|journal=Medico-Legal Journal of Sri Lanka|language=en|volume=5|issue=1|doi=10.4038/mljsl.v5i1.7351|issn=2012-8231}}&lt;/ref&gt;.

Modern-day empirical studies have attested to the usefulness of religion in mitigating stress and elevating mental health&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-94-007-4276-5_22|title=Religion and Mental Health|last=Schieman|first=Scott|last2=Bierman|first2=Alex|last3=Ellison|first3=Christopher G.|date=2013|publisher=Springer Netherlands|isbn=978-94-007-4276-5|editor-last=Aneshensel|editor-first=Carol S.|location=Dordrecht|pages=457–478|language=en|doi=10.1007/978-94-007-4276-5_22|editor-last2=Phelan|editor-first2=Jo C.|editor-last3=Bierman|editor-first3=Alex}}&lt;/ref&gt;. Religion has been found to be positively correlated with improved mental health, and more religious patients were concluded to have &quot;better mental health and adapt[ed] more quickly to health problems&quot; versus patients who weren't religious&lt;ref&gt;{{Cite journal|last=Koenig|first=Harold G.|date=2012|title=Religion, spirituality, and health: the research and clinical implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3671693/|journal=ISRN psychiatry|volume=2012|pages=278730|doi=10.5402/2012/278730|issn=2090-7966|pmc=3671693|pmid=23762764}}&lt;/ref&gt;. [https://www.researchgate.net/scientific-contributions/T-N-Wickramarathna-2247724082 Dr. Wickramarathna] of the University Psychiatry Unit (UPU) at the National Hospital of Sri Lanka (NHSL) argues that psychiatrists must strive for a balance in their approach to patients and &quot;make positive use of religion in [their] practice[s]&quot;&lt;ref&gt;{{Cite journal|last=Wickramarathna|first=T. N.|date=2022-12-31|title=Psychiatrists should stand far from the shrine: why and why not we should separate religion from psychiatry|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v13i2.8397|journal=Sri Lanka Journal of Psychiatry|language=en|volume=13|issue=2|doi=10.4038/sljpsyc.v13i2.8397|issn=2012-6883}}&lt;/ref&gt;.

=== Buddhism ===
27 Sinhalese Buddhists from four Buddhist temples were selected for a series of 70-minute interviews and focus group discussions with the aim of learning the Sinhala Buddhist understanding and experience of spiritual well-being and psychological well-being. The interviewees held spiritual wellness to be the &quot;center&quot; of overall wellness, the &quot;precondition for a successful life&quot;&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/articles/10.4038/sljss.v44i1.7990|journal=Sri Lanka Journal of Social Sciences|language=en-US|volume=44|issue=1|doi=10.4038/sljss.v44i1.7990|issn=0258-9710}}&lt;/ref&gt;. Sinhala Buddhists believe that wellness cannot be achieved without spiritual tranquility. The report states that participants emphasized that spirituality &quot;cannot be directly intervened&quot; and can only be seen through &quot;[interactions] with society&quot;&lt;ref name=&quot;:14&quot; /&gt;. Despite the ''athmaya'' (soul) being &quot;unreachable&quot;, it can be &quot;intervened&quot;, or treated, through the actions of the mind and body with society&lt;ref name=&quot;:14&quot; /&gt;. One being &quot;psychologically ill&quot; can affect one's spiritual being, as the participants reported in their interviews, and can be impaired through &quot;lifestyle stressors, environmental and socio-cultural causes, non-human related causes and bad-karma in the past lives&quot;&lt;ref name=&quot;:14&quot; /&gt;.  

The researchers concluded that despite Sinhala Buddhists not being able to articulately decipher the discrepancies between psychological well-being and spiritual well-being, they are able to conceptualize and maintain a culturally embedded understanding between the two, serving as reputable evidence of the integration of mental health in Sinhala Buddhist practices. However, it is important to note that these results come from a very small sample size and cannot be generalized to all Sri Lankan Buddhists.   

In addition, a 2009 study found that a belief in karma was correlated with poor health. However, an earlier study found a positive correlation between the reliance on the [[w:Karma_in_Buddhism|Buddhist concept of karma]] and trauma, inferencing Buddhist karma being a prevalent response to trauma&lt;ref&gt;{{Cite journal|last=Levy|first=Becca R.|last2=Slade|first2=Martin D.|last3=Ranasinghe|first3=Padmini|date=2009-03|title=Causal thinking after a tsunami wave: karma beliefs, pessimistic explanatory style and health among Sri Lankan survivors|url=https://pubmed.ncbi.nlm.nih.gov/19229624|journal=Journal of Religion and Health|volume=48|issue=1|pages=38–45|doi=10.1007/s10943-008-9162-5|issn=1573-6571|pmid=19229624}}&lt;/ref&gt;. Overall, the effectiveness of karma as a coping mechanism appears to be conflicted.   

Studies indicate that other practices of Buddhism seem to be utilized by individuals affected by the war. 40% of Sri Lankan Buddhists affected by the 2004 tsunami found the Buddhist ritual ''Bodhipuja'' to be helpful in dealing with traumatic experiences&lt;ref&gt;{{Cite web|url=https://jmvh.org/article/mental-health-and-the-role-of-cultural-and-religious-support-in-the-assistance-of-disabled-veterans-in-sri-lanka/|title=Mental Health and the Role of Cultural and Religious Support in the Assistance of Disabled Veterans in Sri Lanka|website=JMVH|language=en-US|access-date=2025-08-12}}&lt;/ref&gt;.    

=== Catholicism ===
Catholic counseling refers to &quot;a nuanced and holistic mental health care paradigm that intricately weaves together psychological science with the moral, spiritual, and pastoral traditions of the Catholic Church&quot;&lt;ref name=&quot;:13&quot;&gt;Perera, U. [https://www.researchgate.net/profile/Udeshini-Perera/publication/394095042_Catholic_Counselling_in_Sri_Lanka_Integrating_Faith_Psychology_and_Cultural_Healing/links/6889303af8031739e6098c79/Catholic-Counselling-in-Sri-Lanka-Integrating-Faith-Psychology-and-Cultural-Healing.pdf Catholic Counselling in Sri Lanka: Integrating Faith, Psychology, and Cultural Healing]. July 2025.&lt;/ref&gt; and aims to assimilate Catholic theology and evidence-based psychological treatment while including Sri Lankan cultural elements. This is achieved through emphasis on community cohesion and a locally-based understanding of &quot;personhood&quot;&lt;ref name=&quot;:13&quot; /&gt;. 

The origins of Catholic counseling trace back to the introduction of Roman Catholicism to the island in the 1600s, with the focus of the early Sri Lankan Catholic community being on &quot;[[w:Evangelism|evangelization]], education, and sacramental formation&quot;. Demand for counseling services in general increased due to the impacts of the Sri Lankan Civil War, where Catholic organizations (Caritas Sri Lanka, Seth Sarana, Subodhi Integral Centre (Piliyandala), etc.) established several Catholic-based trauma-informed programmes for victims of the Civil War. Programmes use group therapy, forgiveness rituals, and narrative repairs to alleviate war trauma. 

Examples of integration of Catholic virtues and counseling can be seen in [[w:Cognitive_Behavioral_Therapy|Cognitive Behavioral Therapy]] (CBT), where &quot;hope&quot; and &quot;humility&quot; are used as the frameworks for creating spiritual resilience&lt;ref name=&quot;:13&quot; /&gt;. The general Christian call for &quot;agape love and acceptance&quot; is echoed by the concept of [[w:Unconditional_positive_regard|unconditional positive regard]]. ''[[w:Lectio_Divina|Lectio Divina]]'' (Catholic prayer and meditation) and ''Marian devotions'' are integrated into therapeutic practices to achieve emotional regulation and mindfulness.   

Senior Lecturer [https://www.researchgate.net/profile/Udeshini-Perera Udeshini Perera] of the University of Colombo articulates a critical role of Catholic counseling. She claims that secular counseling fails to address the &quot;spiritual roots of distress and moral confusion&quot;. Catholic counseling fills in this gap by integrating &quot;psychological insights with a transcendent orientation, supporting lasting transformation and integrity&quot;&lt;ref name=&quot;:13&quot; /&gt;.  

As of 2025, no formal accreditation or standardized training exists for [[w:Pastoral_counseling|pastoral counselors]] in Sri Lanka, hampering the legitimacy of Catholic counseling. Udeshini Perera remarks that mental health stigma, lack of standardized training, research regarding Catholic counseling effectiveness, and acceptance of the combination of religion and science in a professional setting present challenges for Catholic pastoral counseling in the country. Additionally, Catholic psychiatry in Sri Lanka appears to be under-researched, and evidence of its empirical effects on followers appears sparse. Further research is needed in assessing the empirical effects of Catholic counseling in Sri Lanka.  

=== Islam ===
The literature on the empirical effects of Islamic-based psychotherapy in Sri Lanka is limited. Research is limited to a 2012 case study of a 21-year-old Muslim woman experiencing episodic possession states. The patient ceased attending psychiatric services and opted for religious rituals. The patient reported, in a follow-up visit, that the possession states had been absent for 3 months since her switch to religious rituals. The woman and her family attributed the apparent improvement of her condition to religious rituals&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|last3=Yoosuf|first3=Alam|last4=Karunaratne|first4=Sanjeewani|last5=de Silva|first5=Pushpa|date=2012|title=Religious Beliefs, Possession States, and Spirits: Three Case Studies from Sri Lanka|url=http://www.hindawi.com/journals/crips/2012/232740/|journal=Case Reports in Psychiatry|language=en|volume=2012|pages=1–3|doi=10.1155/2012/232740|issn=2090-682X|pmc=3437272|pmid=22970398}}&lt;/ref&gt;.

Future recommendations would be to conduct research on the foundations of Islamic psychiatry in the country, and to observe the rituals implemented and their effects on patients. Studies have found that Islamic prayer can be an effective means of &quot;support and coping&quot;&lt;ref name=&quot;:15&quot; /&gt;. Seven world-wide case studies using Islamic-based psychotherapy on patients, consisting of religious rituals such as scriptural reading from the [[w:Quran|Quran]], teaching of fundamental Islamic concepts (such as ''[[w:Tawakkul|tawakkul]]''), and active implementation of contemplation (''[[w:Tadabbur|tadabbur]]''), have reported positive effects in decreasing cognitive and emotional symptoms associated with &quot;religious, obsessive-compulsive disorder, depression, agoraphobia, generalized anxiety disorder, grief, and substance use disorder.”&lt;ref&gt;{{Cite journal|last=Kurhade|first=Chhaya Shantaram|last2=Jagannathan|first2=Aarti|last3=Varambally|first3=Shivarama|last4=Shivanna|first4=Sushrutha|date=2022-01|title=Religion-based interventions for mental health disorders: A systematic review|url=https://journals.lww.com/10.4103/ijoyppp.ijoyppp_14_21|journal=Journal of Applied Consciousness Studies|language=en|volume=10|issue=1|pages=20–33|doi=10.4103/ijoyppp.ijoyppp_14_21|issn=2949-6993}}&lt;/ref&gt; Additionally, a community-based study of elderly patients in Bangalore, India receiving Islamic-based psychotherapy observed decreased exhibitions of sleep disorders, eating disorders, and emotional distress&lt;ref&gt;{{Cite journal|last=Hafeez|first=Nimin|last2=Sanjay|first2=Thittamaranahalli Varadappa|last3=Puthussery|first3=Yannick Poulose|last4=Madhusudan|first4=Muralidhar|last5=Kariyappa|first5=Poornima Muddaiah|last6=Kulkarni|first6=Sridevi|last7=Raj|first7=Lavanya|date=2023-12-31|title=Spiritual practices among elderly, prevalence, pattern and associated factors: a community-based study from rural Bengaluru, India|url=https://jccpsl.sljol.info/articles/10.4038/jccpsl.v29i4.8610|journal=Journal of the College of Community Physicians of Sri Lanka|language=en|volume=29|issue=4|doi=10.4038/jccpsl.v29i4.8610|issn=1391-3174}}&lt;/ref&gt;. 

=== Hinduism ===
Despite Hindus being 12.6% of the population of Sri Lanka, the research on Hinduism-based therapy in the country is limited. Ayurvedic medicine, a form of medicine originating from ancient India, predominated the Sri Lankan medical landscape for over 2,000 years and even had a symbiotic relationship with Sinhalese medicine, which also played a significant and influential role in the country's medical framework&lt;ref name=&quot;:0&quot; /&gt;&lt;ref&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/article/10.4038/sljss.v44i1.7990/|journal=Sri Lanka Journal of Social Sciences|volume=44|issue=1|pages=33|doi=10.4038/sljss.v44i1.7990|issn=2478-1169}}&lt;/ref&gt;. Despite its historical dominance, Ayurvedic medicine has been challenged against modern evidence-based medical standards&lt;ref&gt;{{Cite book|url=https://philarchive.org/rec/DOMAAT|title=Ayurveda: Ancient Tradition or Pseudoscientific Practice? A Philosophical Inquiry|last=Dominic|first=Shubham K.}}&lt;/ref&gt;.

=== Comparative synthesis ===
Taking an overarching review of the role of religion in Sri Lanka, methods to improve mental well-being are practiced by adherents of Buddhism, Hinduism, Islam, and Christianity. These practices are implemented in traditionally-oriented mental health care, which has been reportedly preferred over psychiatric care at times. These rituals practiced across these religions indicate a common theme of psychologically integrated aspects of well-being. Interpretation of trauma is a central use in religion, with religious principles, such as karma and ''tawakkul'', serve as psychologically analogous mechanisms during times of distress.  

In terms of methodological comparisons to the studies described, qualitative interviews have documented Buddhist practices and principles, like Bodhipuja and the belief in karma, in response to traumatic events, while case studies found religious practices by other religious groups, such as a Muslim patient reading Islamic scripture and observing prayer, to reduce emotional distress. Peer-reviewed sources have documented Catholic practices and principles, such as ''Lectio Divina'' and unconditional positive regard, in improving mindfulness and emotional regulation. The paper acknowledges limitations in the evaluation of certain findings, such as in Islam and Hinduism. These shortcomings, however, are a reflection of the existing literature and its deficiencies. Empirical findings indicate mental health practices are complex and are multifaceted in their effects. 

Evidently, religion serves a parallel role to psychiatric services in improving mental health. Despite its perceived benefits, the findings surrounding religions' role in mental health suffer from conflicting, and sometimes contradictory, results. Additionally, a disproportionate amount of empirical findings seem to be Buddhist-predominant, while other religions are underrepresented in the research. Regarding research barriers, the methodological approaches implemented to study the practices of religious followers vary, though much of the research was brought from qualitative or case-based studies, impeding generalizability. Another noteworthy issue is that many studies do not utilize standardized, psychiatric measures.

== Future Outlook ==
Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients&lt;ref name=&quot;:6&quot; /&gt;. Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were &quot;outdated&quot; and descriptions of clinical roles remained ambiguous&lt;ref name=&quot;:16&quot; /&gt;. A draft mental health legislation from 2007 included provisions for human rights, but due to &quot;bureaucratic processes&quot; and a &quot;lack of consensus&quot;, the draft has not been officially approved.

These limitations pose challenges to the standardization of mental healthcare admissions and may impact the rights of detained patients. Detained patients may have their human rights violated due to a lack of updated legal framework, thereby impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}&lt;/ref&gt;. Lastly, current efforts should ideally move beyond just addressing poverty-centered matters, and expand efforts to domestic violence victims and children with disabilities, as shelters and specialized services are limited&lt;ref name=&quot;:82&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.

Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health framework, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. Without the necessary reforms to advance Sri Lankan mental health legislation, clinicians and vulnerable patients may suffer from a lack of comprehensive oversight. 

==Acknowledgements==
No acknowledgments have been made.

==References==
{{reflist|35em}}

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&lt;div style=&quot;text-align: center;&quot;&gt;''Remotely proctored exam during the exam period''&lt;/div&gt;
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==Overview==
* Weight: 40%
* 2-hour, online, remotely proctored exam during the Week 14-15 University exam period
* Multiple-choice and open-ended questions
* 50% about motivation and 50% about emotion
* Assesses knowledge and learning from [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Tutorials|tutorials]], and readings
* Closed-book: Notes and genAI are not permitted
* For more detail, see the [https://uclearn.canberra.edu.au/courses/20143/quizzes/71013 exam guidelines]

==Marking and feedback==
* Marks and feedback will be provided when official university grades are released (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates]).

==Deferred exams==
* Apply for a deferred exam using the University's [https://www.canberra.edu.au/content/myuc/home/my-studies/exams/about-uc-exams.html deferred exam application] with appropriate documentary evidence
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* If you don't sit the exam (or the deferred exam) it is unlikely that you will pass the unit

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
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| Identify the major principles of motivation and emotion
| The exam tests understanding of key theories and research in the field of motivation and emotion as emphasised in the textbook, lectures, and tutorials.
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==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
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| Be professional — up-to-date knowledge and skills
| Develop a broad understanding of current psychological theory and research about motivation and emotion.
|}
&lt;!--
==Instructions==
* '''Attempts''': Unlimited attempts
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==See also==
* [[Motivation and emotion/Assessment/Exam/Practice quizzes|Practice quizzes]]
{{Motivation and emotion/Assessment/Navigation}}
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      <text bytes="3115" sha1="gw8am6gcc1l2dsp9m97pm5s88q66cu0" xml:space="preserve">{{title|Exam — Guidelines}}
&lt;div style=&quot;text-align: center;&quot;&gt;''Remotely proctored exam during the exam period''&lt;/div&gt;
&lt;!-- {{Motivation and emotion/Assessment/In development}} --&gt;
{{TOCright}}
==Overview==
* Weight: 40%
* 2-hour, online, remotely proctored exam during the Week 14-15 University exam period
* Multiple-choice and open-ended questions
* 50% about motivation and 50% about emotion
* Assesses knowledge and learning from [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Tutorials|tutorials]], and readings
* Closed-book: Notes and genAI are not permitted
* For more detail, see the [https://uclearn.canberra.edu.au/courses/20143/quizzes/71013 exam guidelines]

==Marking and feedback==
* Marks and feedback will be provided when official university grades are released (see [https://www.canberra.edu.au/about-uc/key-dates/ Key dates]).

==Deferred exams==
* Apply for a deferred exam using the University's [https://www.canberra.edu.au/content/myuc/home/my-studies/exams/about-uc-exams.html deferred exam application] with appropriate documentary evidence.
* Submissions are accepted up to 3 days late (-10% per day late).
* If you don't sit the exam (or the deferred exam) it is unlikely that you will pass the unit.

==Learning outcomes==
How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Learning outcome'''
| style=&quot;width:60%;&quot; | '''Assessment task'''
|-
| Identify the major principles of motivation and emotion
| The exam tests understanding of key theories and research in the field of motivation and emotion as emphasised in the textbook, lectures, and tutorials.
|}

==Graduate attributes==
How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise:
{| border=1 cellpadding=5 cellspacing=&quot;0&quot; background:transparent style=&quot;width:90%; margin: auto;&quot;
|-
| style=&quot;width:40%;&quot; | '''Graduate attribute'''
| style=&quot;width:60%;&quot; | '''Assessment task''' 
|-
| Be professional — up-to-date knowledge and skills
| Develop a broad understanding of current psychological theory and research about motivation and emotion.
|}
&lt;!--
==Instructions==
* '''Attempts''': Unlimited attempts
* '''Availability''': Quizzes will be available on {{Motivation and emotion/Canvas}} throughout semester
* '''Content''': Quizzes consist of 10 randomly selected multiple-choice questions from a test bank designed to assess knowledge of content covered in the corresponding lectures, tutorials, and readings
* '''Time limit''': 15 minutes
* '''Bug bounty''' (bonus marks): Email the [[Motivation and emotion/About/Staff|unit convener]] if you identify a quiz error or possible improvement. Accepted revisions earn bonus quiz marks.
--&gt;
==See also==
* [[Motivation and emotion/Assessment/Exam/Practice quizzes|Practice quizzes]]
{{Motivation and emotion/Assessment/Navigation}}
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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
    <ns>2</ns>
    <id>326765</id>
    <revision>
      <id>2819144</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 600-cell */</comment>
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      <text bytes="87352" sha1="hip94egxv5nca8tbapvuoar4ha6ptss" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, invariant as they move in 4-space. The isocline chords form the rotation's Clifford polygons, stationary in 4-space, over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|42°
|138°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|52°
|128°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|56°
|124°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|64°
|116°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|68°
|112°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|84°
|96°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="87383" sha1="1j1vrv2haa22c1m94ji7288v411t2oa" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, invariant as their vertices circle in 4-space. The isocline chords form the rotation's Clifford polygons, stationary in 4-space, over which the edge polygons' vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|42°
|138°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|52°
|128°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|56°
|124°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|64°
|116°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|68°
|112°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|84°
|96°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="575" sha1="qybl3zvbsxl08kmjcc96n3h1soxr6wr" xml:space="preserve">Welcome to a dialogical space dedicated to discussions about a 2026 essay titled &quot;''(Re)imagining (an)other wor(l)d(s) through undisciplinary research.'' ''Reflections on epistemic, existential, sustainable and solidarity-based (E2-S2) work''&quot;, published in ''Annual Review of Applied Linguistics'', ARAL 2026 issue on the theme: '''Interdisciplinarity in Applied Linguistics''' (the official journal of the ''American Association for Applied Linguistics''). See Doi: 10.1017/S0267190526100348 and https://www.cambridge.org/core/journals/annual-review-of-applied-linguistics.</text>
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      <text bytes="30381" sha1="ly4fqndlm8qhfs6ebsii3gk8qi8r6xf" xml:space="preserve">:''This discusses a 2026-07-01 interview with Dimitar Zagorski&lt;ref name=Dimi&gt;&lt;!--Dimitar Zagorski, 3rd--&gt;{{cite Q|Q130719781}}&lt;/ref&gt; about the European Commission's targeted public consultation on copyright law, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-06-27 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
[[File:Wikimedia concerns with European copyright rules including AI and scientific research.webm|thumb|2026-07-01 interview of Dimi Zagorski regarding [[meta:Wikimedia Europe|Wikimedia]] concerns with [[w:European Union|European]] [[w:Copyright law of the European Union|copyright]] rules including [[w:Artificial intelligence|AI]] and scientific research]]
[[File:Wikimedia concerns with European copyright rules including AI and scientific research.ogg|thumb|29:00 mm:ss excerpts from a 2026-07-01 interview of Dimi Zagorski regarding [[meta:Wikimedia Europe|Wikimedia]] concerns with [[w:European Union|European]] [[w:Copyright law of the European Union|copyright]] rules including [[w:Artificial intelligence|AI]] and scientific research]]
Dimitar (&quot;Dimi&quot;) Zagorski, 3rd.,&lt;ref name=Dimi/&gt; discusses the European Commission's targeted public consultation on copyright law and other issues that concern the [[w:Wikimedia Foundation|Wikimedia Foundation]] including AI, scientific research, and the statutory review of the 2019 [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]]. Zagorski is Policy Director for Wikimedia Europe in Brussels. He is interviewed by Spencer Graves.&lt;ref&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt;

As background for this interview, we review the structure of the government of the [[w:European Union|European Union]] (EU).

== European Union ==

The European Union (EU) is a political and economic union of 27 member states located primarily in Europe with [[w:Special territories of members of the European Economic Area|32 special territories]] or subnational units of EU member states stretching from [[w:Greenland|Greenland]] to the southern Indian ocean and the South Pacific. The EU has a population of over 450 million and nominal gross domestic product (GDP) of around €19 trillion in 2025; this makes it roughly one sixth of the global economy. The [[w:Institutions of the European Union|Institutions of the European Union]] encompass seven principal decision-making bodies:

# [[w:European Parliament|European Parliament]], whose approval is required for proposed legislation to become law. It currently has 720 members (MEPs). It functions roughly like the [[w:United States House of Representatives|House of Representatives in the US]], in that MEPs can amend or reject proposed legislation, though they cannot initiate legislation. 
# [[w:European Council|European Council]] of heads of state or government.
# [[w:Council of the European Union|Council of the European Union]], often referred to simply as the Council and less formally as the &quot;Council of Ministers&quot;. This is a legislative body, which works with the European Parliament to amend and approve or veto proposals of the European Commission, which holds the right of initiative; neither the Council nor the Parliament can initiate legislation. The Presidency of the council is not a single post, but is held by a member state's government and rotates every six months. It functions roughly like the [[w:United States Senate|Senate in the US]] but cannot initiate legislation. 
# [[w:European Commission|European Commission]] (EC), the executive cabinet of the European Union. Currently, there is one Commissioner per member state, including the president (currently [[w:Ursula von der Leyen|Ursula von der Leyen]]), but members are bound by their oath of office to represent the interest of the EU as a whole rather than their home state. All EU legislation must be initiated by the Commission, to be amended and approved or vetoed by the European Parliament and the Council. 
# [[w:Court of Justice of the European Union|Court of Justice of the European Union]].
# [[w:European Central Bank|European Central Bank]]. 
# [[w:European Court of Auditors|European Court of Auditors]]. 

== Highlights == 

:''These excerpts from the interview are rushed, lightly edited for readability, and may not be in final form. The ultimate authority on what was said is the accompanying video.''

=== Wikimedia Foundation activities and projects === 

Graves began by asking for a brief overview of Wikimedia Foundation activities and projects. Zagorski replied, &quot;The Wikimedia Foundation is the legal host of projects like [[w:Wikipedia|Wikipedia]], [[w:Wikimedia Commons|Wikimedia Commons]], and [[w:Wikidata|Wikidata]]. It also has a lot of affiliates around the world. These are independent organizations, usually taking the form of an association, as [[w:Wikimedia Deutschland|Wikimedia Deutschland]] in Germany, or [[w:Wikimedia Nederland|Wikimedia Netherlands]] in the Netherlands, or [[Wikimedia:Wikimedia Italia|Wikimedia Italy]], a membership association in Italy. The European affiliates of the Wikimedia Foundation have created a joint organization called [[meta:Wikimedia Europe|Wikimedia Europe]], which I'm working for. Otherwise, the community mostly works on assembling and curating knowledge online on projects such as Wikipedia.&quot;

Graves asked, &quot;The Wikimedia Foundation does not pay anyone to edit Wikipedia, right?&quot; Zagorski replied, &quot;That is correct. The Wikimedia movement has decided to not engage in paid editing activities.&quot;

Graves continued, &quot;The Wikimedia Foundation maintains the software, organizes conferences, and lobbies governmental agencies like you do?&quot; Zagorski added, &quot;Very important, they maintain not only the software, they have the physical servers where the projects physically live and where we access them from. The Wikimedia Foundation runs on its own metal. They have their own racks and their own computers that are the servers. This is where we get Wikipedia and its sister projects from.&quot;

Graves then said, &quot;In a recent email to the public policy group for Wikimedia, you mentioned several things, starting with a digital omnibus and open data. Talk about that.&quot; Zagorski replied, {{quote|

In the field of policy making in Brussels, and in Europe in general there are two big major questions that are looming over the internet. 

# One is age verification: Should there be some age verification online or not, and for which platforms? 
# The other one is, oh my god, [[w:Artificial intelligence|artificial intelligence]] (AI) is coming. Are we all gonna die, or what are we going to do now? 

These are the two big overarching political discussions here right now. In the field of AI, one of the big questions that rights holders and publishers are asking is, &quot;Will anybody still see our products? And will we ever get paid?&quot; Because they think that AI essentially uses their works more or less for free, because of copyright exceptions. 

For Wikimedia, we are having a very comprehensive approach to AI. On one hand, Wikimedia content gets used a lot by AI for AI training. That's allowed, because it's an open license. Most of the big AI models use Wikipedia and sister projects to establish what they call &quot;[[w:ground truth|ground truth]]&quot;, which means that they give it a higher weight to a claim coming, let's say, from Wikipedia than from other places. It's not perfect, but it's one of the better places on the Internet. 

There are two things: There is the base training. It's done, I don't know, once every few years, or maybe a bit more often, but not all too often. And then there is the inference. Inference means when you type something into an AI chat and it gives you an answer. 

For us it's fine. For rights holders or publishers, maybe it's not fine. 

But it also uses a lot of [[w:Web crawler|crawlers]] that go to your servers. Wikimedia has experienced a humongous increase of these crawlers, while human readers are down, so less people go to Wikipedia, but a lot more [[w:Web crawler|bots]] come. Our mission is to spread knowledge and free knowledge, so mission aligned. Maybe you could even claim that more people have access to free knowledge, so for us, should be fine. 

But where this puts a real strain is there are several things. First, the infrastructure: It's a lot more expensive to serve bot traffic, because I told you Wikimedia has its own servers. There are a few of them in the US. Then in Europe you have servers in [[w:Amsterdam|Amsterdam]], in [[w:Marseille|Marseille]]. You have one in [[w:Brazil|Brazil]] and one in [[w:Singapore|Singapore]]. And they serve the global traffic.

Bots, like these crawler bots, access a lot of different types of information. It's random, there is no connection between one and the other, so you cannot [[w:Cache (computing)|cache]]. That makes the traffic a lot more expensive. 

Now, the second bit where this is a challenge for us is that Wikipedia and the projects rely on humans actually staying on the project, fixing a mistake, editing, saying, &quot;Okay, this is missing something. We need to add something here now.&quot;  

If people in the future increasingly only use this [[w:Chatbot|chatbot]] search mode -- which has many advantages: It's just very easy to use it. Then even if they improve, let's say, linking, and citing the sources very well -- a lot fewer people would come to Wikipedia. And then we run the risk that actually our community will shrink, and there won't be any people to maintain Wikipedia anymore, which is a major worry that we're having. 

On the other hand, AI tools could be used to make Wikipedia editing maybe easier and more fun, and to take away the tasks that editors need to perform that are really not fun. I mean, vandalism, patrolling, and so on. A lot of this can be helped with AI. So there is both an opportunity, and the risk, of course, 

And the third thing I wanted to also mention is currently the vast majority of income that of donations that Wikimedia gets is from banners on Wikipedia. If fewer people go to the website, to wikipedia.org they won't see the fundraising banners, and we might be getting fewer donations, which would make us less independent, which is a worry we have. 

We are trying to see this as a bit of an opportunity. Or at least we're not waiting for a government to come along and fix this for us. We're trying to fix things ourselves, which I'm quite proud of as a movement. 

There is something called [[wikimedia:enterprise|Wikimedia Enterprise]]. Wikipedia content is free and free to reuse, also for commercial purposes. We don't want to limit this. However, these crawlers by large AI systems put such a toll on our infrastructure that we also need to figure out how to do this. So how do you at the same time keep it open and free for everyone to reuse but try to get at least some money back from these very large players? Wikimedia Enterprise has the following solution: Wikimedia Enterprise essentially offers a very high speed direct access to the content: It's an [[w:API|API]]. So imagine it's like a direct pipe to the to all the content, and there is a guaranteed uptime of, I don't know, 99.999% and all these things that commercial, large scale commercial reusers really want. And then basically they pay for this direct high-speed access rather than for the content itself. 

This might work for other projects, like [[w:OpenStreetMap|OpenStreetMap]] and [[w:Project Gutenberg|Project Gutenberg]]. It probably won't work for everybody on the Internet. 

So there is a big question. How do we make the economics work without locking everybody into these large AI companies and killing off everybody who is smaller? 

=== Weaponized AI === 

One thing that we're still not seeing, but that we are wary of is the weaponized use of AI to manipulate biases on Wikipedia and its sister projects.  

There are a few people who have been caught using AI, but it was obvious and blatant. And then different language versions.  On Wikipedia every language version has its own rules, since every community creates its own rules. 

If you're a malicious actor, suddenly with one person, you could probably imitate being 20 or 30 users or even more. We haven't really seen this, but this is something we are looking out for.}}

=== Combatting vandalism on Wikipedia === 

Graves said, &quot;I remember hearing at one point that IP addresses for the United States Congress were blocked. Do you have a comment on that?&quot; Zagorski agreed, &quot;This happens for institutions.&quot; 

Graves noted, &quot;I get several emails a day saying that a Wikipedia article that I'm &quot;watching&quot; has been changed. So I check. That's at least one way that vandalism is counteracted, right?&quot; Zagorski replied, &quot;Yes. If there are a lot of unconstructive contributions from a certain IP address, it might be blocked, and in this case it obviously happened.&quot; 

=== Child sexual abuse material (CSAM) === 

Graves then recalled, &quot;In your recent policy email, you also talked about [[w:Child pornography|child sexual abuse material (CSAM)]]. You want to talk about that? Zagorski replied, {{quote|

No, but we have to. There is a proposal in Europe that platforms be required to scan their content for child sexual abuse materials. There will be a center in the Netherlands, a European center that basically gathers and creates hashes, so creates fingerprints of known child sexual abuse material, or likely child sexual abuse material. And then you use this to scan through the servers. 

This is already happening in the US. It's called NCMEC, [[w:National Center for Missing &amp; Exploited Children|National Center for Missing &amp; Exploited Children]]. They have such a hash database. The Wikimedia Foundation actually already scans its content using Microsoft's [[w:PhotoDNA|PhotoDNA]] technology to identify such and then report to NCMEC in Washington, DC. 

Something very similar will happen in Europe, and that's okay. We don't mind that. Of course this should be done. The big question mark -- and this has no direct impact on Wikimedia projects, but it has an impact on our users, potentially -- is that one of the proposals is that the scanning for such child sexual abuse material should also include scanning private messages -- instant messaging like WhatsApp, Viber, Signal, and so on. The worry there is that in order to do this, you would have to break end to end encryption.}} 

=== Age verification === 

Graves then asked, &quot;You mentioned age verification online. Talk about that.&quot; Zagorski replied, {{quote| 

This is a global discussion from Australia. Every single EU country, the US, several US states (I think) are discussing this. I think the conversation in Latin America and some African countries is also picking up. So this is really global. 

It's taking us a while. Finally, politicians now are running around and screaming, &quot;Oh my god, who will think of the children?&quot;  

Quite often what gets proposed is a very concrete solution. Australia was the first to do it: Let's have a fixed ban, 16 and under cannot be on social media. 

There are two parts to it. We already require some sort of age verification in real life:  like if somebody wants to buy alcohol or tobacco. 

I understand that there needs to be such a system also on the internet.&lt;ref&gt;Robert Corn-Revere, Chief Counsel for the Foundation for Individual Rights and Expression (FIRE) questions the need for age verification on the Internet, as discussed in the section below on &quot;[[#Questioning the need for age verification|Questioning the need for age verification]]&quot;.&lt;/ref&gt; But the question is, which services or which places on the Internet do we want to be behind such a barrier? And we're essentially going around and saying, &quot;It's not necessary and would not be proportional to put Wikipedia behind an age verification wall. For us this has several arguments:  

First, we want knowledge to be open and free, and even the simplest age verification wall would mean that there is still an extra barrier to people accessing knowledge. 

Secondly, we really care about what they call data minimization. The Wikimedia Foundation tries to collect as little information about its users as possible, because we believe in privacy. We believe that also privacy is very important to intellectual freedom, to be able to read and research whatever you want without the feeling that somebody's watching. 

Mostly we've been successful because we already have a law on the books in France. There are some discussions in Italy and in Greece. And in Sweden there is a legal proposal. 

Mostly, they go after social media, and they define social media in a way that Wikipedia and its sister projects are not included. That is actually the tricky part, because quite often when policymakers create legislation in order to have obligations on Facebook and on Google, they write it quite often in a way that Wikipedia gets caught in the same net, although they never meant for Wikipedia to get caught.}} 

=== The Wikipedia Test ===

Zagorski continued, 'The Wikimedia Foundation has created something called &quot;[[Wikimedia:The Wikipedia Test|The Wikipedia Test]]&quot;: When you're thinking about new legislation or a new rule on the books, you should stop for a moment and think, &quot;Will this law negatively affect independent community-led non-for-profit platforms?&quot; If it negatively impacts Wikipedia, it will likely negatively impact all sorts of independent community-led projects. If a law passes The Wikipedia Test, it means it is safe for independent communities online, and that's a good thing.'

=== Research on age verification? === 

Graves asked, &quot;On age verification, who do you think has the best research proposals on that?&quot; Zagorski said, &quot;A study was published a couple of days ago looking at Australia's ban. I don't know whether it has the desired effect. It might also be early. But what do we want to achieve? Do we want for kids to be safer online? And then I don't know. A ban might not reduce the risks online. It might even increase them.&quot;&lt;ref&gt;[[Foundation for Individual Rights and Expression|Robert Corn-Revere, who was interviewed for this series 2026-07-02]], noted that the US Surgeon General (2023) said that social media is beneficial for some young people and harmful for others: Social media help marginalized kids feel like they're not alone. It pushes others to suicide. He recommends social media literacy training for teachers and parents. He opposes actions that do not limit [[First Amendment to the United States Constitution|First Amendment rights]].&lt;/ref&gt; 

=== Review of the 2019 Directive on Copyright in the Single Digital Market (CDSM) ===

Graves asked, &quot;Summarize for us Wikimedia's contributions to the current European review of the 2019 [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Single Digital Market]] (CDSM). Zagorski replied, {{quote|

This is the last major copyright reform that we had in Europe. It got passed in 2019, and it's written in the law itself that the commission has to do a review after a few years. That's starting now. As part of this, the Commission has issued a call for evidence to seek input. One of their questions was on AI and copyright. How is it working? 

Most notably, there are text and data mining exceptions in the EU that allow anybody to mine publicly accessible content and to use it for, let's say, AI training, unless the owner of the website or the service has explicitly, in a machine-readable way, said, &quot;No, we don't like to be mined now.&quot;  

European rights holders and publishers are complaining that this opt-out is not respected by large LLMs ([[w:Large language model|Large Language Models]]). So there is quite some tension. 

Wikimedia Europe participates in this call for evidence. We mentioned Wikimedia Enterprise is something we do. We might need to better safeguard the commons in the public domain.  

On AI and copyright, we essentially said that the culture of share alike is not always being well respected, so there are a few very large organizations that sort of seem to be sucking up all the information, but it's questionable how much they give back. 

And I don't necessarily mean financial giving back, but also linking back, and making sure that they don't break things and are good citizens.}}

=== Scientific research ===  

{{quote|Another part of this consultation asked about scientific research. There is a copyright exception for scientific research in Europe, and we took a position that this exception that was introduced in 2019 does not work very well cross border for us. It's very important that access to scientific research is as open and free as possible. We need this to write better Wikipedia articles. And it would be important that if you read the Wikipedia article and there is a citation that you can really click on the citation and check for yourself whether the link actually says what Wikipedia claims it does. Otherwise you just need to trust people. So it makes everything less valuable. So we want the cross border element to be improved.  

Another thing we are demanding is something that is called a &quot;secondary publication right for publicly funded research&quot;.&lt;ref&gt;Zagorski (2025).&lt;/ref&gt; The issue there is the following: In Europe, most universities are actually public and publicly funded. And then the taxpayer, in the end, pays for these universities, pays for the research that the professors do in these universities. And then these professors, in order to have a good academic career, need to publish in specific journals that are, of course, important, and that have a history. But publishing in these journals quite often costs money. So, from the public money, we paid this publisher to have our publicly funded works published there.

And then these journals are not open. That costs a lot of money. So you need to again pay for the university library to have access to this journal. And then still the public doesn't have access to this research that we funded four times over by now.  

So our suggestion is to allow researchers who got paid with public money a secondary publication right: Let's say there is a Journal of Medicine, and you published your great article there. You got all your academic scores. And you can say, &quot;Hey, I've been published there&quot;, which is very important also for funding when you work in academia. 

But then after this, maybe immediately, maybe after a few months, you should also be allowed to publish it under a free license online, so anybody can access it. 

Six countries in the EU already have this, including France, Germany, Bulgaria. We believe that this is something that should be applied across the EU.&lt;ref&gt;The US does not have a &quot;copyright exception for scientific research&quot;, like Europe does, as discussed in the [[#Discussion|discussion]] section below in [[#Paying multiple times for scientific research|paying multiple times for scientific research]].&lt;/ref&gt;}}

=== Communia Association === 

Graves then noted, &quot;You mentioned that Wikimedia Europe is a member of the [[w:Communia|Communia Association]] and supports its submissions to this consultation. What's the Communia Association?&quot; Zagorski replied, {{quote|

The Community Association is a group of people and organizations who want to defend and grow the public domain. ... Everybody wants more exclusive rights, and that shrinks the public domain. 

We are in there with other organizations. [[w:Creative Commons|Creative Commons]] is a member. Wikimedia Europe is a member. The [[w:Internet Archive|Internet Archive]] is a member. It's like an umbrella organization for free knowledge and free license people and organizations.}}

Graves noted that they had 20 policy recommendations and asked if Zagorski would care to highlight or summarize a few. Zagorski replied, &quot;Yes, on the part of the educational and the scientific research exception, we're basically in lockstep with Communia there. They really focus on this cross-border access. The 2019 directive was supposed to create a digital single market, but we see that there are so many carve-outs and exceptions written into this law that essentially it might be very hard for you to access public research in Sweden if you're in Malta, or vice versa. Cross-border access to academic content is crucial both for the public, but also for researchers.&quot; 

=== Mastodon and PeerTube === 

Graves said, &quot;I read that the European Union between 2022 and 2024 actually had launched their own [[w:Mastodon (social network)|Mastodon]] and [[w:PeerTube|PeerTube]] instances.&quot; Zagorski replied, &quot;Yes. That's okay. Launching a Mastodon instance is not like creating a new network yourself. It's just the way Mastodon works: You can host your users on your own server.&quot; 

=== In sum === 

Graves noted they were about out of time and asked for final words for the audience. Zagorski replied, {{quote|

It's an interesting time. AI is not just another trendy fashion. We as a society need to decide what we want and how we want it. There are some very honest conversations we need to have, and I hope they won't be ideological, but just let's be pragmatic and talk about what we can and should do.&lt;ref&gt; 

Acemoglu and Johnson ...

&lt;/ref&gt; }}

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

=== Questioning the need for age verification === 

Robert Corn-Revere, Chief Counsel for the Foundation for Individual Rights and Expression (FIRE) seems to disagree with Zargorski's comments above on &quot;[[#Age verification|Age verification]]&quot;: Corn-Revere said that schools can provide cell phone lockers and forbid students from bringing their cell phones into the classroom. He also said that parents can forbid their children from taking their cell phones to bed. And both parents and teachers can help children learn about predators of all kinds -- sexual, financial -- on the Internet. For more see the [[Foundation for Individual Rights and Expression|2026-07-02 interview with Corn-Revere in this series]].

=== Paying multiple times for scientific research ===  

Zagorski's comments above on the European &quot;[[#Scientific research|copyright exception for scientific research]]&quot; does not apply in the US. Graves (2023) expressed concern about this, asking, &quot;Do copyrights and paywalls on academic journals violate the US Constitution?&quot; He noted that the [[w:Copyright Clause|Copyright Clause]] of the US Constitution gives the US Congress the power &quot;To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.&quot; He strongly suggests that copyrights and paywalls on refereed academic journals are obstacles to progress of science and the useful arts in violation of the US Constitution, because no one writes for refereed academic journals while expecting financial rewards from copyright royalties. 

Similarly, Baker (2016, ch. 5) noted that &quot;The Bayh-Dole Act of 1980 allowed for universities, research institutions, private companies, and individuals operating on government contracts to gain control of patents derived from their work, thereby creating the opportunity for universities to earn large rents from patents and for researchers to form their own companies, all relying on knowledge and expertise obtained on government contracts.&quot; 

Graves and Baker seem to suggest that these features of US law seem to violate the principle of free publication employed in some European countries.  

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Dean Baker (2016)  Rigged: How globalization and the rules to the modern economy were structured to make the rich richer--&gt;{{cite Q|Q100216001}}
* &lt;!--Spencer Graves (2023-03-21) &quot;Do copyrights and paywalls on academic journals violate the US Constitution?&quot;--&gt;{{cite Q|Q126116187}}
* &lt;!--U. S. Surgeon General (2023-05-23) Social Media and Youth Mental Health: The U.S. Surgeon General’s Advisory--&gt;{{cite Q|Q118978725|author=U. S. Surgeon General}}
* &lt;!-- Dimitar Zagorski (2025-10-23)&quot;Why Wikimedia Supports Secondary Publication Rights for Publicly Funded Research&quot;--&gt;{{cite Q|Q140650182}}

[[Category:Media]]
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[[Category:Democracy]]
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      <text bytes="31126" sha1="1ka8bpu609idjevox7d0cpfttge8ffx" xml:space="preserve">:''This discusses a 2026-07-01 interview with Dimitar Zagorski&lt;ref name=Dimi&gt;&lt;!--Dimitar Zagorski, 3rd--&gt;{{cite Q|Q130719781}}&lt;/ref&gt; about the European Commission's targeted public consultation on copyright law, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-06-27 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
[[File:Wikimedia concerns with European copyright rules including AI and scientific research.webm|thumb|2026-07-01 interview of Dimi Zagorski regarding [[meta:Wikimedia Europe|Wikimedia]] concerns with [[w:European Union|European]] [[w:Copyright law of the European Union|copyright]] rules including [[w:Artificial intelligence|AI]] and scientific research]]
[[File:Wikimedia concerns with European copyright rules including AI and scientific research.ogg|thumb|29:00 mm:ss excerpts from a 2026-07-01 interview of Dimi Zagorski regarding [[meta:Wikimedia Europe|Wikimedia]] concerns with [[w:European Union|European]] [[w:Copyright law of the European Union|copyright]] rules including [[w:Artificial intelligence|AI]] and scientific research]]
Dimitar (&quot;Dimi&quot;) Zagorski, 3rd.,&lt;ref name=Dimi/&gt; discusses the European Commission's targeted public consultation on copyright law and other issues that concern the [[w:Wikimedia Foundation|Wikimedia Foundation]] including AI, scientific research, and the statutory review of the 2019 [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]]. Zagorski is Policy Director for Wikimedia Europe in Brussels. He is interviewed by Spencer Graves.&lt;ref&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt;

As background for this interview, we review the structure of the government of the [[w:European Union|European Union]] (EU).

== European Union ==

The European Union (EU) is a political and economic union of 27 member states located primarily in Europe with [[w:Special territories of members of the European Economic Area|32 special territories]] or subnational units of EU member states stretching from [[w:Greenland|Greenland]] to the southern Indian ocean and the South Pacific. The EU has a population of over 450 million and nominal gross domestic product (GDP) of around €19 trillion in 2025; this makes it roughly one sixth of the global economy. The [[w:Institutions of the European Union|Institutions of the European Union]] encompass seven principal decision-making bodies:

# [[w:European Parliament|European Parliament]], whose approval is required for proposed legislation to become law. It currently has 720 members (MEPs). It functions roughly like the [[w:United States House of Representatives|House of Representatives in the US]], in that MEPs can amend or reject proposed legislation, though they cannot initiate legislation. 
# [[w:European Council|European Council]] of heads of state or government.
# [[w:Council of the European Union|Council of the European Union]], often referred to simply as the Council and less formally as the &quot;Council of Ministers&quot;. This is a legislative body, which works with the European Parliament to amend and approve or veto proposals of the European Commission, which holds the right of initiative; neither the Council nor the Parliament can initiate legislation. The Presidency of the council is not a single post, but is held by a member state's government and rotates every six months. It functions roughly like the [[w:United States Senate|Senate in the US]] but cannot initiate legislation. 
# [[w:European Commission|European Commission]] (EC), the executive cabinet of the European Union. Currently, there is one Commissioner per member state, including the president (currently [[w:Ursula von der Leyen|Ursula von der Leyen]]), but members are bound by their oath of office to represent the interest of the EU as a whole rather than their home state. All EU legislation must be initiated by the Commission, to be amended and approved or vetoed by the European Parliament and the Council. 
# [[w:Court of Justice of the European Union|Court of Justice of the European Union]].
# [[w:European Central Bank|European Central Bank]]. 
# [[w:European Court of Auditors|European Court of Auditors]]. 

== Highlights == 

:''These excerpts from the interview are rushed, lightly edited for readability, and may not be in final form. The ultimate authority on what was said is the accompanying video.''

=== Wikimedia Foundation activities and projects === 

Graves began by asking for a brief overview of Wikimedia Foundation activities and projects. Zagorski replied, &quot;The Wikimedia Foundation is the legal host of projects like [[w:Wikipedia|Wikipedia]], [[w:Wikimedia Commons|Wikimedia Commons]], and [[w:Wikidata|Wikidata]]. It also has a lot of affiliates around the world. These are independent organizations, usually taking the form of an association, as [[w:Wikimedia Deutschland|Wikimedia Deutschland]] in Germany, or [[w:Wikimedia Nederland|Wikimedia Netherlands]] in the Netherlands, or [[Wikimedia:Wikimedia Italia|Wikimedia Italy]], a membership association in Italy. The European affiliates of the Wikimedia Foundation have created a joint organization called [[meta:Wikimedia Europe|Wikimedia Europe]], which I'm working for. Otherwise, the community mostly works on assembling and curating knowledge online on projects such as Wikipedia.&quot;

Graves asked, &quot;The Wikimedia Foundation does not pay anyone to edit Wikipedia, right?&quot; Zagorski replied, &quot;That is correct. The Wikimedia movement has decided to not engage in paid editing activities.&quot;

Graves continued, &quot;The Wikimedia Foundation maintains the software, organizes conferences, and lobbies governmental agencies like you do?&quot; Zagorski added, &quot;Very important, they maintain not only the software, they have the physical servers where the projects physically live and where we access them from. The Wikimedia Foundation runs on its own metal. They have their own racks and their own computers that are the servers. This is where we get Wikipedia and its sister projects from.&quot;

Graves then said, &quot;In a recent email to the public policy group for Wikimedia, you mentioned several things, starting with a digital omnibus and open data. Talk about that.&quot; Zagorski replied, {{quote|

In the field of policy making in Brussels, and in Europe in general there are two big major questions that are looming over the internet. 

# One is age verification: Should there be some age verification online or not, and for which platforms? 
# The other one is, oh my god, [[w:Artificial intelligence|artificial intelligence]] (AI) is coming. Are we all gonna die, or what are we going to do now? 

These are the two big overarching political discussions here right now. In the field of AI, one of the big questions that rights holders and publishers are asking is, &quot;Will anybody still see our products? And will we ever get paid?&quot; Because they think that AI essentially uses their works more or less for free, because of copyright exceptions. 

For Wikimedia, we are having a very comprehensive approach to AI. On one hand, Wikimedia content gets used a lot by AI for AI training. That's allowed, because it's an open license. Most of the big AI models use Wikipedia and sister projects to establish what they call &quot;[[w:ground truth|ground truth]]&quot;, which means that they give it a higher weight to a claim coming, let's say, from Wikipedia than from other places. It's not perfect, but it's one of the better places on the Internet. 

There are two things: There is the base training. It's done, I don't know, once every few years, or maybe a bit more often, but not all too often. And then there is the inference. Inference means when you type something into an AI chat and it gives you an answer. 

For us it's fine. For rights holders or publishers, maybe it's not fine. 

But it also uses a lot of [[w:Web crawler|crawlers]] that go to your servers. Wikimedia has experienced a humongous increase of these crawlers, while human readers are down, so less people go to Wikipedia, but a lot more [[w:Web crawler|bots]] come. Our mission is to spread knowledge and free knowledge, so mission aligned. Maybe you could even claim that more people have access to free knowledge, so for us, should be fine. 

But where this puts a real strain is there are several things. First, the infrastructure: It's a lot more expensive to serve bot traffic, because I told you Wikimedia has its own servers. There are a few of them in the US. Then in Europe you have servers in [[w:Amsterdam|Amsterdam]], in [[w:Marseille|Marseille]]. You have one in [[w:Brazil|Brazil]] and one in [[w:Singapore|Singapore]]. And they serve the global traffic.

Bots, like these crawler bots, access a lot of different types of information. It's random, there is no connection between one and the other, so you cannot [[w:Cache (computing)|cache]]. That makes the traffic a lot more expensive. 

Now, the second bit where this is a challenge for us is that Wikipedia and the projects rely on humans actually staying on the project, fixing a mistake, editing, saying, &quot;Okay, this is missing something. We need to add something here now.&quot;  

If people in the future increasingly only use this [[w:Chatbot|chatbot]] search mode -- which has many advantages: It's just very easy to use it. Then even if they improve, let's say, linking, and citing the sources very well -- a lot fewer people would come to Wikipedia. And then we run the risk that actually our community will shrink, and there won't be any people to maintain Wikipedia anymore, which is a major worry that we're having. 

On the other hand, AI tools could be used to make Wikipedia editing maybe easier and more fun, and to take away the tasks that editors need to perform that are really not fun. I mean, vandalism, patrolling, and so on. A lot of this can be helped with AI. So there is both an opportunity, and the risk, of course, 

And the third thing I wanted to also mention is currently the vast majority of income that of donations that Wikimedia gets is from banners on Wikipedia. If fewer people go to the website, to wikipedia.org they won't see the fundraising banners, and we might be getting fewer donations, which would make us less independent, which is a worry we have. 

We are trying to see this as a bit of an opportunity. Or at least we're not waiting for a government to come along and fix this for us. We're trying to fix things ourselves, which I'm quite proud of as a movement. 

There is something called [[wikimedia:enterprise|Wikimedia Enterprise]]. Wikipedia content is free and free to reuse, also for commercial purposes. We don't want to limit this. However, these crawlers by large AI systems put such a toll on our infrastructure that we also need to figure out how to do this. So how do you at the same time keep it open and free for everyone to reuse but try to get at least some money back from these very large players? Wikimedia Enterprise has the following solution: Wikimedia Enterprise essentially offers a very high speed direct access to the content: It's an [[w:API|API]]. So imagine it's like a direct pipe to the to all the content, and there is a guaranteed uptime of, I don't know, 99.999% and all these things that commercial, large scale commercial reusers really want. And then basically they pay for this direct high-speed access rather than for the content itself. 

This might work for other projects, like [[w:OpenStreetMap|OpenStreetMap]] and [[w:Project Gutenberg|Project Gutenberg]]. It probably won't work for everybody on the Internet. 

So there is a big question. How do we make the economics work without locking everybody into these large AI companies and killing off everybody who is smaller? 

=== Weaponized AI === 

One thing that we're still not seeing, but that we are wary of is the weaponized use of AI to manipulate biases on Wikipedia and its sister projects.  

There are a few people who have been caught using AI, but it was obvious and blatant. And then different language versions.  On Wikipedia every language version has its own rules, since every community creates its own rules. 

If you're a malicious actor, suddenly with one person, you could probably imitate being 20 or 30 users or even more. We haven't really seen this, but this is something we are looking out for.}}

=== Combatting vandalism on Wikipedia === 

Graves said, &quot;I remember hearing at one point that IP addresses for the United States Congress were blocked. Do you have a comment on that?&quot; Zagorski agreed, &quot;This happens for institutions.&quot; 

Graves noted, &quot;I get several emails a day saying that a Wikipedia article that I'm &quot;watching&quot; has been changed. So I check. That's at least one way that vandalism is counteracted, right?&quot; Zagorski replied, &quot;Yes. If there are a lot of unconstructive contributions from a certain IP address, it might be blocked, and in this case it obviously happened.&quot; 

=== Child sexual abuse material (CSAM) === 

Graves then recalled, &quot;In your recent policy email, you also talked about [[w:Child pornography|child sexual abuse material (CSAM)]]. You want to talk about that? Zagorski replied, {{quote|

No, but we have to. There is a proposal in Europe that platforms be required to scan their content for child sexual abuse materials. There will be a center in the Netherlands, a European center that basically gathers and creates hashes, so creates fingerprints of known child sexual abuse material, or likely child sexual abuse material. And then you use this to scan through the servers. 

This is already happening in the US. It's called NCMEC, [[w:National Center for Missing &amp; Exploited Children|National Center for Missing &amp; Exploited Children]]. They have such a hash database. The Wikimedia Foundation actually already scans its content using Microsoft's [[w:PhotoDNA|PhotoDNA]] technology to identify such and then report to NCMEC in Washington, DC. 

Something very similar will happen in Europe, and that's okay. We don't mind that. Of course this should be done. The big question mark -- and this has no direct impact on Wikimedia projects, but it has an impact on our users, potentially -- is that one of the proposals is that the scanning for such child sexual abuse material should also include scanning private messages -- instant messaging like WhatsApp, Viber, Signal, and so on. The worry there is that in order to do this, you would have to break end to end encryption.}} 

=== Age verification === 

Graves then asked, &quot;You mentioned age verification online. Talk about that.&quot; Zagorski replied, {{quote| 

This is a global discussion from Australia. Every single EU country, the US, several US states (I think) are discussing this. I think the conversation in Latin America and some African countries is also picking up. So this is really global. 

It's taking us a while. Finally, politicians now are running around and screaming, &quot;Oh my god, who will think of the children?&quot;  

Quite often what gets proposed is a very concrete solution. Australia was the first to do it: Let's have a fixed ban, 16 and under cannot be on social media. 

There are two parts to it. We already require some sort of age verification in real life:  like if somebody wants to buy alcohol or tobacco. 

I understand that there needs to be such a system also on the internet.&lt;ref&gt;Robert Corn-Revere, Chief Counsel for the Foundation for Individual Rights and Expression (FIRE) questions the need for age verification on the Internet, as discussed in the section below on &quot;[[#Questioning the need for age verification|Questioning the need for age verification]]&quot;.&lt;/ref&gt; But the question is, which services or which places on the Internet do we want to be behind such a barrier? And we're essentially going around and saying, &quot;It's not necessary and would not be proportional to put Wikipedia behind an age verification wall. For us this has several arguments:  

First, we want knowledge to be open and free, and even the simplest age verification wall would mean that there is still an extra barrier to people accessing knowledge. 

Secondly, we really care about what they call data minimization. The Wikimedia Foundation tries to collect as little information about its users as possible, because we believe in privacy. We believe that also privacy is very important to intellectual freedom, to be able to read and research whatever you want without the feeling that somebody's watching. 

Mostly we've been successful because we already have a law on the books in France. There are some discussions in Italy and in Greece. And in Sweden there is a legal proposal. 

Mostly, they go after social media, and they define social media in a way that Wikipedia and its sister projects are not included. That is actually the tricky part, because quite often when policymakers create legislation in order to have obligations on Facebook and on Google, they write it quite often in a way that Wikipedia gets caught in the same net, although they never meant for Wikipedia to get caught.}} 

=== The Wikipedia Test ===

Zagorski continued, 'The Wikimedia Foundation has created something called &quot;[[Wikimedia:The Wikipedia Test|The Wikipedia Test]]&quot;: When you're thinking about new legislation or a new rule on the books, you should stop for a moment and think, &quot;Will this law negatively affect independent community-led non-for-profit platforms?&quot; If it negatively impacts Wikipedia, it will likely negatively impact all sorts of independent community-led projects. If a law passes The Wikipedia Test, it means it is safe for independent communities online, and that's a good thing.'

=== Research on age verification? === 

Graves asked, &quot;On age verification, who do you think has the best research proposals on that?&quot; Zagorski said, &quot;A study was published a couple of days ago looking at Australia's ban. I don't know whether it has the desired effect. It might also be early. But what do we want to achieve? Do we want for kids to be safer online? And then I don't know. A ban might not reduce the risks online. It might even increase them.&quot;&lt;ref&gt;[[Foundation for Individual Rights and Expression|Robert Corn-Revere, who was interviewed for this series 2026-07-02]], noted that the US Surgeon General (2023) said that social media is beneficial for some young people and harmful for others: Social media help marginalized kids feel like they're not alone. It pushes others to suicide. He recommends social media literacy training for teachers and parents. He opposes actions that do not limit [[First Amendment to the United States Constitution|First Amendment rights]].&lt;/ref&gt; 

=== Review of the 2019 Directive on Copyright in the Single Digital Market (CDSM) ===

Graves asked, &quot;Summarize for us Wikimedia's contributions to the current European review of the 2019 [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Single Digital Market]] (CDSM). Zagorski replied, {{quote|

This is the last major copyright reform that we had in Europe. It got passed in 2019, and it's written in the law itself that the commission has to do a review after a few years. That's starting now. As part of this, the Commission has issued a call for evidence to seek input. One of their questions was on AI and copyright. How is it working? 

Most notably, there are text and data mining exceptions in the EU that allow anybody to mine publicly accessible content and to use it for, let's say, AI training, unless the owner of the website or the service has explicitly, in a machine-readable way, said, &quot;No, we don't like to be mined now.&quot;  

European rights holders and publishers are complaining that this opt-out is not respected by large LLMs ([[w:Large language model|Large Language Models]]). So there is quite some tension. 

Wikimedia Europe participates in this call for evidence. We mentioned Wikimedia Enterprise is something we do. We might need to better safeguard the commons in the public domain.  

On AI and copyright, we essentially said that the culture of share alike is not always being well respected, so there are a few very large organizations that sort of seem to be sucking up all the information, but it's questionable how much they give back. 

And I don't necessarily mean financial giving back, but also linking back, and making sure that they don't break things and are good citizens.}}

=== Scientific research ===  

{{quote|Another part of this consultation asked about scientific research. There is a copyright exception for scientific research in Europe, and we took a position that this exception that was introduced in 2019 does not work very well cross border for us. It's very important that access to scientific research is as open and free as possible. We need this to write better Wikipedia articles. And it would be important that if you read the Wikipedia article and there is a citation that you can really click on the citation and check for yourself whether the link actually says what Wikipedia claims it does. Otherwise you just need to trust people. So it makes everything less valuable. So we want the cross border element to be improved.  

Another thing we are demanding is something that is called a &quot;secondary publication right for publicly funded research&quot;.&lt;ref&gt;Zagorski (2025).&lt;/ref&gt; The issue there is the following: In Europe, most universities are actually public and publicly funded. And then the taxpayer, in the end, pays for these universities, pays for the research that the professors do in these universities. And then these professors, in order to have a good academic career, need to publish in specific journals that are, of course, important, and that have a history. But publishing in these journals quite often costs money. So, from the public money, we paid this publisher to have our publicly funded works published there.

And then these journals are not open. That costs a lot of money. So you need to again pay for the university library to have access to this journal. And then still the public doesn't have access to this research that we funded four times over by now.  

So our suggestion is to allow researchers who got paid with public money a secondary publication right: Let's say there is a Journal of Medicine, and you published your great article there. You got all your academic scores. And you can say, &quot;Hey, I've been published there&quot;, which is very important also for funding when you work in academia. 

But then after this, maybe immediately, maybe after a few months, you should also be allowed to publish it under a free license online, so anybody can access it. 

Six countries in the EU already have this, including France, Germany, Bulgaria. We believe that this is something that should be applied across the EU.&lt;ref&gt;The US does not have a &quot;copyright exception for scientific research&quot;, like Europe does, as discussed in the [[#Discussion|discussion]] section below in [[#Paying multiple times for scientific research|paying multiple times for scientific research]].&lt;/ref&gt;}}

=== Communia Association === 

Graves then noted, &quot;You mentioned that Wikimedia Europe is a member of the [[w:Communia|Communia Association]] and supports its submissions to this consultation. What's the Communia Association?&quot; Zagorski replied, {{quote|

The Community Association is a group of people and organizations who want to defend and grow the public domain. ... Everybody wants more exclusive rights, and that shrinks the public domain. 

We are in there with other organizations. [[w:Creative Commons|Creative Commons]] is a member. Wikimedia Europe is a member. The [[w:Internet Archive|Internet Archive]] is a member. It's like an umbrella organization for free knowledge and free license people and organizations.}}

Graves noted that they had 20 policy recommendations and asked if Zagorski would care to highlight or summarize a few. Zagorski replied, &quot;Yes, on the part of the educational and the scientific research exception, we're basically in lockstep with Communia there. They really focus on this cross-border access. The 2019 directive was supposed to create a digital single market, but we see that there are so many carve-outs and exceptions written into this law that essentially it might be very hard for you to access public research in Sweden if you're in Malta, or vice versa. Cross-border access to academic content is crucial both for the public, but also for researchers.&quot; 

=== Mastodon and PeerTube === 

Graves said, &quot;I read that the European Union between 2022 and 2024 actually had launched their own [[w:Mastodon (social network)|Mastodon]] and [[w:PeerTube|PeerTube]] instances.&quot; Zagorski replied, &quot;Yes. That's okay. Launching a Mastodon instance is not like creating a new network yourself. It's just the way Mastodon works: You can host your users on your own server.&quot; 

=== In sum === 

Graves noted they were about out of time and asked for final words for the audience. Zagorski replied, {{quote|

It's an interesting time. AI is not just another trendy fashion. We as a society need to decide what we want and how we want it. There are some very honest conversations we need to have, and I hope they won't be ideological, but just let's be pragmatic and talk about what we can and should do.&lt;ref&gt; 

Acemoglu and Johnson ...

&lt;/ref&gt; }}

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

=== Questioning the need for age verification === 

Robert Corn-Revere, Chief Counsel for the Foundation for Individual Rights and Expression (FIRE) seems to disagree with Zargorski's comments above on &quot;[[#Age verification|Age verification]]&quot;: Corn-Revere said that schools can provide cell phone lockers and forbid students from bringing their cell phones into the classroom. He also said that parents can forbid their children from taking their cell phones to bed. And both parents and teachers can help children learn about predators of all kinds -- sexual, financial -- on the Internet. For more see the [[Foundation for Individual Rights and Expression|2026-07-02 interview with Corn-Revere in this series]].

In this vein, [[Let's agree to disagree and seek common ground|Nolan Higdon, interviewed 2026-05-29 for this series]], and others are working on proposals for teaching media literacy and AI literacy. He expressed concern about &quot;How Corporatism Paved the Way for the Hostile Takeover of Higher Ed&quot;.&lt;ref&gt;e.g., Higdon (2006).&lt;/ref&gt; He and others are working on a curriculum for teachers interested in teaching AI literacy, including content that major corporations might prefer ''not'' be taught, because it could help students (and their parents) make better decisions to the detriment of major corporations. 

=== Paying multiple times for scientific research ===  

Zagorski's comments above on the European &quot;[[#Scientific research|copyright exception for scientific research]]&quot; does not apply in the US. Graves (2023) expressed concern about this, asking, &quot;Do copyrights and paywalls on academic journals violate the US Constitution?&quot; He noted that the [[w:Copyright Clause|Copyright Clause]] of the US Constitution gives the US Congress the power &quot;To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.&quot; He strongly suggests that copyrights and paywalls on refereed academic journals are obstacles to progress of science and the useful arts in violation of the US Constitution, because no one writes for refereed academic journals while expecting financial rewards from copyright royalties. 

Similarly, Baker (2016, ch. 5) noted that &quot;The Bayh-Dole Act of 1980 allowed for universities, research institutions, private companies, and individuals operating on government contracts to gain control of patents derived from their work, thereby creating the opportunity for universities to earn large rents from patents and for researchers to form their own companies, all relying on knowledge and expertise obtained on government contracts.&quot; 

Graves and Baker seem to suggest that these features of US law seem to violate the principle of free publication employed in some European countries.  

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Dean Baker (2016)  Rigged: How globalization and the rules to the modern economy were structured to make the rich richer--&gt;{{cite Q|Q100216001}}
* &lt;!--Spencer Graves (2023-03-21) &quot;Do copyrights and paywalls on academic journals violate the US Constitution?&quot;--&gt;{{cite Q|Q126116187}}
* &lt;!--Nolan Higdon (2026-03-09) MAGAcademy: How Corporatism Paved the Way for the Hostile Takeover of Higher Ed--&gt;{{cite Q|Q139897231}}
* &lt;!--U. S. Surgeon General (2023-05-23) Social Media and Youth Mental Health: The U.S. Surgeon General’s Advisory--&gt;{{cite Q|Q118978725|author=U. S. Surgeon General}}
* &lt;!-- Dimitar Zagorski (2025-10-23)&quot;Why Wikimedia Supports Secondary Publication Rights for Publicly Funded Research&quot;--&gt;{{cite Q|Q140650182}}

[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Media literacy]]
[[Category:Wikimedia]]
[[Category:Wikimedia Foundation]]
[[Category:Wikimedia Foundation staff]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
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    <title>Probability Dilation Theory / Matrix Dilation Operators</title>
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      <text bytes="4000" sha1="krwuyekd44h3r1ehbibxpna0zbgzrt5" xml:space="preserve">= Probability Dilation Theory / Matrix Dilation Operators =

== Overview ==

This page explores a natural mathematical generalization of the Probability Dilation Theory (PDT) operator by replacing the diagonal dilation matrix with a general nonnegative matrix. The resulting framework allows interactions between probability states and extends localized probability dilation to coupled probability evolution.

This page is exploratory and proposes a possible extension of PDT rather than a replacement for the core theory.

== Motivation ==

The standard PDT transformation applies a positive diagonal dilation operator

&lt;math&gt;
T_D(P)=\frac{DP}{Z(P,D)},
&lt;/math&gt;

where

&lt;math&gt;
Z(P,D)=\sum_i D_iP_i,
&lt;/math&gt;

and

&lt;math&gt;
D=
\begin{pmatrix}
d_1&amp;0&amp;\cdots&amp;0\\
0&amp;d_2&amp;\cdots&amp;0\\
\vdots&amp;\vdots&amp;\ddots&amp;\vdots\\
0&amp;0&amp;\cdots&amp;d_n
\end{pmatrix}.
&lt;/math&gt;

Each probability component is independently reweighted before normalization.

A natural question is whether the diagonal operator can be generalized to a full matrix while preserving positivity and normalization.

== Generalized Matrix Operator ==

Consider a nonnegative matrix

&lt;math&gt;
M=
\begin{pmatrix}
d_1&amp;a_{12}&amp;\cdots&amp;a_{1n}\\
a_{21}&amp;d_2&amp;\cdots&amp;a_{2n}\\
\vdots&amp;\vdots&amp;\ddots&amp;\vdots\\
a_{n1}&amp;a_{n2}&amp;\cdots&amp;d_n
\end{pmatrix},
&lt;/math&gt;

where

&lt;math&gt;
M_{ij}\ge0.
&lt;/math&gt;

The generalized PDT operator is defined by

&lt;math&gt;
T_M(P)
=
\frac{MP}{\|MP\|_1},
&lt;/math&gt;

provided that

&lt;math&gt;
MP
&lt;/math&gt;

has nonnegative components.

Normalization ensures that

&lt;math&gt;
\sum_i T_M(P)_i=1.
&lt;/math&gt;

== Interpretation ==

The diagonal entries

&lt;math&gt;
d_i
&lt;/math&gt;

represent localized probability dilation identical to standard PDT.

The off-diagonal entries

&lt;math&gt;
a_{ij}
&lt;/math&gt;

allow probability associated with one state to influence another.

Consequently,

* diagonal PDT represents independent local amplification;
* matrix PDT represents coupled probability evolution.

== Relation to Standard PDT ==

When all off-diagonal entries vanish,

&lt;math&gt;
a_{ij}=0,\qquad i\ne j,
&lt;/math&gt;

the generalized operator reduces exactly to the original PDT operator.

Thus diagonal PDT is a special case of matrix PDT.

== Connections to Existing Mathematics ==

The generalized formulation exhibits similarities to several established mathematical frameworks, including

* positive matrices,
* Markov operators,
* Perron–Frobenius theory,
* coupled dynamical systems,
* graph-based diffusion,
* linear operator theory.

These connections are mathematical analogies and are not claimed to be equivalent formulations.

== Potential Properties ==

Questions for future investigation include:

* preservation of positivity,
* convergence of repeated iterations,
* existence and uniqueness of fixed points,
* entropy evolution,
* stability of coupled systems,
* spectral properties of the dilation matrix,
* relation to Perron–Frobenius eigenvectors,
* continuous-time matrix dilation flows.

== Discussion ==

Matrix Dilation Operators extend the localized dilation concept of PDT by allowing interactions between probability states.

This generalization significantly enlarges the mathematical scope of PDT while preserving the original theory as the diagonal special case.

Whether this extension provides useful applications or deeper theoretical insights remains an open research question.

== See also ==

* [[Probability Dilation Theory]]
* [[Probability Dilation Theory/Measure-Theoretic Foundations]]
* [[Probability Dilation Theory/Dilation Vector Field]]
* [[Probability Dilation Theory/Dilation Flows]]
* [[Probability Dilation Theory/Stochastic Dilation Fields]]
* [[Probability Dilation Theory/Convergence and Fixed Points]]
* [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]]

== References ==

* Howard Richardson, ''Probability Dilation Theory'', Wikiversity.
* R. A. Horn and C. R. Johnson, ''Matrix Analysis''.
* F. R. Gantmacher, ''The Theory of Matrices''.
* O. Perron (1907); G. Frobenius (1912), classical papers on positive matrices.</text>
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      <text bytes="1030" sha1="2ihlzd3zw112jpvzbvk2ujun2r3dxsn" xml:space="preserve">==Afrotropical [[w:Philanthidae|Philanthidae]]==

Philanthidae is a family of stinging wasps (Infraorder [[African Arthropods/Aculeata|Aculeata]]).&lt;ref&gt;https://www.waspweb.org/Apoidea/index.htm&lt;/ref&gt;

Afrotropical Philanthidae comprise three genera in three subfamilies&lt;ref&gt;https://www.waspweb.org/Apoidea/Philanthidae/Classification/index.htm&lt;/ref&gt; (illustrated below).

=== Subfamily Cercerinae ===
*Genus ''Cerceris''
&lt;gallery mode=packed heights=200&gt;
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

=== Subfamily Philanthinae ===
*Genus ''Philanthus''
** [[African_Arthropods/Philanthus South African species of ''Philanthus'']]
&lt;gallery mode=packed heights=200&gt;
Philanthus triangulum diadema 187037342.jpg
African Beewolf (Philanthus triangulum diadema) on Sleepy Morning (Waltheria indica) ... (52730621858).jpg
&lt;/gallery&gt;

=== Subfamily Pseudoscolinae ===
*Genus ''Pseudoscolia''

Only one Afrotropical species. ''Pseudoscolia aequatoria'' Prentice and Pulawski (2004) was found in Kenya.


==References==
{{reflist}}

{{BookCat}}</text>
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      <text bytes="1011" sha1="el8k92ahe65x4zuetxsfkflll6wl4hp" xml:space="preserve">==Afrotropical [[w:Philanthidae|Philanthidae]]==

Philanthidae is a family of stinging wasps (Infraorder [[African Arthropods/Aculeata|Aculeata]]).&lt;ref&gt;https://www.waspweb.org/Apoidea/index.htm&lt;/ref&gt;

Afrotropical Philanthidae comprise three genera in three subfamilies&lt;ref&gt;https://www.waspweb.org/Apoidea/Philanthidae/Classification/index.htm&lt;/ref&gt; (illustrated below).

=== Subfamily Cercerinae ===
*Genus ''Cerceris''
&lt;gallery mode=packed heights=200&gt;
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

=== Subfamily Philanthinae ===
*Genus ''Philanthus''
** [[Philanthus South African species of ''Philanthus'']]
&lt;gallery mode=packed heights=200&gt;
Philanthus triangulum diadema 187037342.jpg
African Beewolf (Philanthus triangulum diadema) on Sleepy Morning (Waltheria indica) ... (52730621858).jpg
&lt;/gallery&gt;

=== Subfamily Pseudoscolinae ===
*Genus ''Pseudoscolia''

Only one Afrotropical species. ''Pseudoscolia aequatoria'' Prentice and Pulawski (2004) was found in Kenya.


==References==
{{reflist}}

{{BookCat}}</text>
      <sha1>el8k92ahe65x4zuetxsfkflll6wl4hp</sha1>
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    <revision>
      <id>2819133</id>
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      <timestamp>2026-07-23T12:13:29Z</timestamp>
      <contributor>
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        <id>1669821</id>
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      <text bytes="1062" sha1="39lqol5wwq2sm9oqolp2xuqhhe4ec84" xml:space="preserve">==Afrotropical [[w:Philanthidae|Philanthidae]]==

Philanthidae is a family of stinging wasps (Infraorder [[African Arthropods/Aculeata|Aculeata]]).&lt;ref&gt;https://www.waspweb.org/Apoidea/index.htm&lt;/ref&gt;

Afrotropical Philanthidae comprise three genera in three subfamilies&lt;ref&gt;https://www.waspweb.org/Apoidea/Philanthidae/Classification/index.htm&lt;/ref&gt; (illustrated below).

=== Subfamily Cercerinae ===
*Genus ''Cerceris''
&lt;gallery mode=packed heights=200&gt;
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

=== Subfamily Philanthinae ===
*Genus ''Philanthus''
** [[https://en.wikiversity.org/wiki/African_Arthropods/Philanthus South African species of ''Philanthus'']]
&lt;gallery mode=packed heights=200&gt;
Philanthus triangulum diadema 187037342.jpg
African Beewolf (Philanthus triangulum diadema) on Sleepy Morning (Waltheria indica) ... (52730621858).jpg
&lt;/gallery&gt;

=== Subfamily Pseudoscolinae ===
*Genus ''Pseudoscolia''

Only one Afrotropical species. ''Pseudoscolia aequatoria'' Prentice and Pulawski (2004) was found in Kenya.


==References==
{{reflist}}

{{BookCat}}</text>
      <sha1>39lqol5wwq2sm9oqolp2xuqhhe4ec84</sha1>
    </revision>
    <revision>
      <id>2819134</id>
      <parentid>2819133</parentid>
      <timestamp>2026-07-23T12:13:48Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Subfamily Philanthinae */</comment>
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      <text bytes="1060" sha1="hdni0rpswz5gdcxr8a3r9rki8ldsdhi" xml:space="preserve">==Afrotropical [[w:Philanthidae|Philanthidae]]==

Philanthidae is a family of stinging wasps (Infraorder [[African Arthropods/Aculeata|Aculeata]]).&lt;ref&gt;https://www.waspweb.org/Apoidea/index.htm&lt;/ref&gt;

Afrotropical Philanthidae comprise three genera in three subfamilies&lt;ref&gt;https://www.waspweb.org/Apoidea/Philanthidae/Classification/index.htm&lt;/ref&gt; (illustrated below).

=== Subfamily Cercerinae ===
*Genus ''Cerceris''
&lt;gallery mode=packed heights=200&gt;
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

=== Subfamily Philanthinae ===
*Genus ''Philanthus''
** [https://en.wikiversity.org/wiki/African_Arthropods/Philanthus South African species of ''Philanthus'']
&lt;gallery mode=packed heights=200&gt;
Philanthus triangulum diadema 187037342.jpg
African Beewolf (Philanthus triangulum diadema) on Sleepy Morning (Waltheria indica) ... (52730621858).jpg
&lt;/gallery&gt;

=== Subfamily Pseudoscolinae ===
*Genus ''Pseudoscolia''

Only one Afrotropical species. ''Pseudoscolia aequatoria'' Prentice and Pulawski (2004) was found in Kenya.


==References==
{{reflist}}

{{BookCat}}</text>
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== Licensing ==
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|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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== Licensing ==
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        <id>2955086</id>
      </contributor>
      <comment>created a sample course</comment>
      <origin>2819146</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="214" sha1="ickzesxpmolx9ep56z7fzch61v26c8b" xml:space="preserve">== My Courses ==

=== Computing for Senior High School in Ghana ===

==== Computing for Senior High School one ====

==== Computing for Senior High School two ====


==== Computing for Senior High School three ====</text>
      <sha1>ickzesxpmolx9ep56z7fzch61v26c8b</sha1>
    </revision>
    <revision>
      <id>2819147</id>
      <parentid>2819146</parentid>
      <timestamp>2026-07-23T15:02:10Z</timestamp>
      <contributor>
        <username>Dnshitobu</username>
        <id>2955086</id>
      </contributor>
      <comment>updated</comment>
      <origin>2819147</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="256" sha1="gaasv8ezwz7hjqdl2dq1fmjl30uyvcw" xml:space="preserve">== My Courses ==

=== [[Computing for Senior High School in Ghana]] ===

==== [[Computing for Senior High School in Ghana/one]] ====

==== [[Computing for Senior High School in Ghana/two]] ====

==== [[Computing for Senior High School in Ghana/three]] ====</text>
      <sha1>gaasv8ezwz7hjqdl2dq1fmjl30uyvcw</sha1>
    </revision>
  </page>
  <page>
    <title>File:Python.Work2.Library.1A.20260723.pdf</title>
    <ns>6</ns>
    <id>330710</id>
    <revision>
      <id>2819160</id>
      <timestamp>2026-07-23T16:42:59Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Work2.1A: Libraries (20260723 - 20260722)
|Source={{own|Young1lim}}
|Date=2026-07-23
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2819160</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="264" sha1="smrigjiytvz5gynxc7mojrznysxbc10" xml:space="preserve">== Summary ==
{{Information
|Description=Work2.1A: Libraries (20260723 - 20260722)
|Source={{own|Young1lim}}
|Date=2026-07-23
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>smrigjiytvz5gynxc7mojrznysxbc10</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Dnshitobu</title>
    <ns>3</ns>
    <id>330711</id>
    <revision>
      <id>2819187</id>
      <timestamp>2026-07-24T00:25:51Z</timestamp>
      <contributor>
        <username>Solstag</username>
        <id>64708</id>
      </contributor>
      <comment>/* Welcome! */ new section</comment>
      <origin>2819187</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="226" sha1="7r42y1wbmlmil20xo1lyyco5uejnjvg" xml:space="preserve">== Welcome! ==

Ni! Welcome @[[User:Dnshitobu|Dnshitobu]], let me know if I can lend you a hand. [[User:Solstag|Solstag]] ([[User talk:Solstag|discuss]] • [[Special:Contributions/Solstag|contribs]]) 00:25, 24 July 2026 (UTC)</text>
      <sha1>7r42y1wbmlmil20xo1lyyco5uejnjvg</sha1>
    </revision>
  </page>
  <page>
    <title>User:ChanghunShin</title>
    <ns>2</ns>
    <id>330712</id>
    <revision>
      <id>2819188</id>
      <timestamp>2026-07-24T01:49:20Z</timestamp>
      <contributor>
        <username>ChanghunShin</username>
        <id>3102925</id>
      </contributor>
      <comment>Create user page</comment>
      <origin>2819188</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="270" sha1="pgx7rg4o3dvo3t93vmmrthxt9kpw46z" xml:space="preserve">
= Changhun Shin =
Independent researcher and creator of '''CS-NRRM™ (Changhun Shin Natural Recovery Pattern Model)'''.

== Research interests ==

* Longitudinal Human Observation
* Continuity-Based Structural Observation
* AI-Readable Longitudinal Data Infrastructure</text>
      <sha1>pgx7rg4o3dvo3t93vmmrthxt9kpw46z</sha1>
    </revision>
    <revision>
      <id>2819189</id>
      <parentid>2819188</parentid>
      <timestamp>2026-07-24T01:50:23Z</timestamp>
      <contributor>
        <username>ChanghunShin</username>
        <id>3102925</id>
      </contributor>
      <comment>Create user page</comment>
      <origin>2819189</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="445" sha1="7e5mydkakphytr8afs2mshii7v2ikti" xml:space="preserve">
= Changhun Shin =
Independent researcher and creator of '''CS-NRRM™ (Changhun Shin Natural Recovery Pattern Model)'''.

== Research interests ==

* Longitudinal Human Observation
* Continuity-Based Structural Observation
* AI-Readable Longitudinal Data Infrastructure

== Official resources ==

* Official website: https://www.cs-nrrm.com
* GitHub: https://github.com/changhunshin-csnrrm/cs-nrrm
* ORCID: https://orcid.org/0009-0001-3805-3023</text>
      <sha1>7e5mydkakphytr8afs2mshii7v2ikti</sha1>
    </revision>
  </page>
  <page>
    <title>User:ChanghunShin/Sandbox</title>
    <ns>2</ns>
    <id>330713</id>
    <revision>
      <id>2819190</id>
      <timestamp>2026-07-24T01:59:36Z</timestamp>
      <contributor>
        <username>ChanghunShin</username>
        <id>3102925</id>
      </contributor>
      <comment>Create personal sandbox</comment>
      <origin>2819190</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="359" sha1="568xcsrjqyvpdt36876ds6g95d3ay5d" xml:space="preserve">= Sandbox =

This is my personal sandbox for developing Wikiversity learning resources.

== Future learning resources ==

* CS-NRRM™
* Longitudinal Human Observation
* Continuity-Based Structural Observation
* AI-Readable Longitudinal Data Infrastructure

== Notes ==

This page is used to draft and organize future educational materials before publication.</text>
      <sha1>568xcsrjqyvpdt36876ds6g95d3ay5d</sha1>
    </revision>
    <revision>
      <id>2819192</id>
      <parentid>2819190</parentid>
      <timestamp>2026-07-24T02:13:21Z</timestamp>
      <contributor>
        <username>ChanghunShin</username>
        <id>3102925</id>
      </contributor>
      <comment>Update sandbox draft</comment>
      <origin>2819192</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="936" sha1="luikyh49zt63eeedwtusl48l4kamvdb" xml:space="preserve">= CS-NRRM™ =

== Introduction ==
CS-NRRM™ (Changhun Shin Natural Recovery Pattern Model) is a non-medical structural observation framework designed to organize longitudinal human observational records while preserving continuity, chronology, metadata, and contextual relationships.

This sandbox is used to develop a future learning resource about the CS-NRRM™ framework.

== Learning outline ==

* Background
* Core Principles
* Continuity-Based Structural Observation
* Chronology
* Metadata
* AI-Readable Longitudinal Data Infrastructure
* References

----

== Previous Sandbox Notes ==
This is my personal sandbox for developing future educational resources.

=== Future learning resources ===

* CS-NRRM™
* Longitudinal Human Observation
* Continuity-Based Structural Observation
* AI-Readable Longitudinal Data Infrastructure

=== Notes ===
This page is used to draft and organize educational materials before publication.</text>
      <sha1>luikyh49zt63eeedwtusl48l4kamvdb</sha1>
    </revision>
  </page>
  <page>
    <title>File:NM.NLE.2Newton.20260721.pdf</title>
    <ns>6</ns>
    <id>330714</id>
    <revision>
      <id>2819216</id>
      <timestamp>2026-07-24T05:51:14Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=2. Newton-Raphson Method (20260721 - 20260720)
|Source={{own|Young1lim}}
|Date=2026-07-24
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2819216</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="265" sha1="tb00m23l61jpm81411z8jkrtvwv41be" xml:space="preserve">== Summary ==
{{Information
|Description=2. Newton-Raphson Method (20260721 - 20260720)
|Source={{own|Young1lim}}
|Date=2026-07-24
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>tb00m23l61jpm81411z8jkrtvwv41be</sha1>
    </revision>
  </page>
  <page>
    <title>File:NM.NLE.2Newton.20260722.pdf</title>
    <ns>6</ns>
    <id>330715</id>
    <revision>
      <id>2819218</id>
      <timestamp>2026-07-24T05:51:59Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=2. Newton-Raphson Method (20260722 - 20260721)
|Source={{own|Young1lim}}
|Date=2026-07-24
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2819218</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="265" sha1="i8mbnq77xw2qbjvzz4dzveardhgb8uq" xml:space="preserve">== Summary ==
{{Information
|Description=2. Newton-Raphson Method (20260722 - 20260721)
|Source={{own|Young1lim}}
|Date=2026-07-24
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>i8mbnq77xw2qbjvzz4dzveardhgb8uq</sha1>
    </revision>
  </page>
  <page>
    <title>Algebra 1/Arithmetic</title>
    <ns>0</ns>
    <id>330716</id>
    <revision>
      <id>2819239</id>
      <timestamp>2026-07-24T10:36:06Z</timestamp>
      <contributor>
        <username>Evan Mercer</username>
        <id>3071189</id>
      </contributor>
      <comment>Moved here</comment>
      <origin>2819239</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17995" sha1="st9lt2hrezspo7axxbru4cw03dxha8p" xml:space="preserve">==Arithmetic==
[[File:Multiply 4 bags 3 marbles.svg|thumb|right|4 x 3 = 12 (multiplication)]]
'''Arithmetic''' has to deal with elementary/basic levels of math, such as division, multiplication, subtraction, and addition. Basically, just working with numbers.

This SHOULD be a level familiar with you. If you are not familiar with arithmetic math/rules, then PLEASE review through Arithmetic, as you won't survive even the 1st step of Algebra. Trust me, the basics are THAT important.
===Fractions===
[[File:Cake quarters.svg|thumb|left|A Cake with fractions]]
'''Fractions''' (from Latin ''fractus'', &quot;broken&quot;) are parts of a whole. On the left side in the image of the cake, there is only &lt;math&gt;3/4&lt;/math&gt;'s of the cake showing, the other &lt;math&gt;1/4&lt;/math&gt; has been eaten/taken away.

The number, ''3'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''numerator''' (Numerator: Number at the top, tells us of how much of the number is being talked about/being used). The number, ''4'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''denominator''' (Denominator: Number showing the all time total).
;
;Simplest form/reduced form
A reduced form of a fraction is a fraction that cannot be divided by any number other than 1, and the denominator is greater than 1. So &lt;math&gt;2/4&lt;/math&gt; is NOT in simplest form, since we can divide 2 and 4, by 2... which results in the following number: &lt;math&gt;1/2&lt;/math&gt;. Though, not every fraction can be divided by 2, there are fractions, such as: &lt;math&gt;5/35&lt;/math&gt;, &lt;math&gt;7/21&lt;/math&gt;, and &lt;math&gt;30/5&lt;/math&gt;. The two first fractions are not divisible by 2, and &lt;math&gt;30/5&lt;/math&gt; can not be divided by 2 on both sides, but only on &lt;math&gt;30&lt;/math&gt;. It's important to simplify as if you were in a test, your teacher will mark your problems as incorrect if you didn't simplify your fractions. Keep in mind that simplifying a fraction into its simplest/reduced form doesn't change its value, both the original (unsimplified) fraction and its reduced form represent the same exact value/quantity. So, &lt;math&gt;\tfrac{2}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{2}&lt;/math&gt; represents the same quantity, a half! 

Here, we will present a few fractions for you to simplify.
====Sample problems for ''simplifying fractions'' (use ''/'' as the fraction line)====
&lt;quiz display=simple points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{6}{8}=&lt;/math&gt;{ 3/4_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{4}{60}=&lt;/math&gt;{ 1/15_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{30}{90}=&lt;/math&gt;{ 1/3_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{8}{18}=&lt;/math&gt;{ 4/9_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{9}{72}=&lt;/math&gt;{ 1/8_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{64}{46}=&lt;/math&gt;{ 32/23|1 9/23_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{206}{340}=&lt;/math&gt;{ 103/170_7 }
&lt;/quiz&gt;
===== Adding or Subtracting Fractions =====
[[File:Fractionsworkalgebra.PNG|thumb|right|What we just worked on, summarized]]
To simply add or subtract fractions, make sure the denominators of the fractions you are adding or subtracting are the same. If they are not, find the least common denominator (LCD). For example, if you want to add &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; and &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, you first have to multiply the 2 in &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; by '''3''', which equals '''6'''... BUT you cannot just multiply 2 only, you also have to multiply 4 by 3, since that's what you did to 2, the denominator. If you change the denominator, you have to change the numerator. ('''This step is crucial as it allows you to preserve the same value of the fraction''' but with just a different representation)

Alright, we got that out of the way, so once we have &lt;math&gt;\tfrac{12}{6}&lt;/math&gt; + &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, we can simply add. So &lt;math&gt;12&lt;/math&gt; + &lt;math&gt;4&lt;/math&gt; = &lt;math&gt;16&lt;/math&gt;, but don't add the denominators, they stay the same. So the answer is &lt;math&gt;\tfrac{16}{6}&lt;/math&gt;, and then we simplify down to &lt;math&gt;\tfrac{8}{3}&lt;/math&gt; dividing by 2 on both the numerator and denominator. 

But... did you notice something? &lt;math&gt;\tfrac{8}{3}&lt;/math&gt;? That doesn't seem right, does it? The denominator is smaller than the numerator. When you have a fraction like this, you have to convert it to a '''mixed fraction''' (skip to [[Speak_Math_Now!/Week_1:_Introduction_To_Algebra#Improper_Fraction_--.3E_Mixed_Fraction|section 2.1.1.4]]).

===== Multiplying Fractions =====
To multiply fractions, its easiest to first simplify your fraction to simplest terms. Once you have done that, you can simply multiply the numerators and the denominators. And obviously, simplify your final product, if you can. So, we have &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; and &lt;math&gt;\tfrac{2}{6}&lt;/math&gt;. You could multiply the numerators and denominators straight away and simplify at the end if you are comfortable, but to make it easier and clearer, we should simplify the fractions first. We simplify 6 and 8 by dividing both by 2, we also divide 2 and 6 by 2. So the fractions are now &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{3}&lt;/math&gt;. You simply multiply those two fractions by multiplying the numerator by the numerator, and doing the same for the denominators. After completing this process, you will get a solution (in fraction form). &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; × &lt;math&gt;\tfrac{1}{3}&lt;/math&gt; &lt;math&gt;=&lt;/math&gt; &lt;math&gt;\tfrac{3}{12}&lt;/math&gt;. 

&lt;math&gt;\tfrac{3}{12}&lt;/math&gt; is not going to be our final product, though, since we can simplify the fraction by dividing the fraction by 3, which results in &lt;math&gt;\tfrac{1}{4}&lt;/math&gt;.

===== Dividing Fractions =====
There is an interesting twist when it comes to dividing fractions. You have to turn the fraction you want to divide by (second fraction) upside-down, also known as &quot;Keep, Change, Flip&quot; where you keep the first fraction the same, change the operation to multiplication, and replace the second fractions numerator with the denominator and the denominator with the numerator. Not only that, you have to turn the division symbol (÷) into a multiplication symbol (× or •). After that, you use your skills you learned in multiplying a fraction, and you multiply both of the fractions. Simplify if you need to.

So, &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; ÷ &lt;math&gt;\tfrac{7}{12}&lt;/math&gt;. Change the division symbol to a multiplication symbol, and turn the fraction you want to divide by upside-down (the upside-down fraction is known as a '''reciprocal'''). So &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; × (or •) &lt;math&gt;\tfrac{12}{7}&lt;/math&gt;. Multiply the numerators and denominators. The answer is &lt;math&gt;\tfrac{72}{56}&lt;/math&gt;, simplified down to &lt;math&gt;\tfrac{9}{7}&lt;/math&gt;.

===== Improper Fraction --&gt; Mixed Fraction =====
Divide the numerator by the denominator. The '''quotient''' (result of the division taking place/number above the division line) will be the whole number of the mixed fraction, while the numerator will be the remainder. The denominator remains unchanged, so don't change the denominator at all!

{{notice|If you would like to take the quiz on Fractions, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Fractions Quiz]]'''}}

See also: https://www.tes.com/lessons/bJieZ4sFPJbSTw/fractions-4-mixed-numbers-and-improper-fractions

===Decimals===

Ever wondered how to write 8&lt;math&gt;\tfrac{47}{100}&lt;/math&gt; as a decimal? Well, you've got the answer: 8.47! How did we get that answer? Let's look at a few more and maybe you'll see the pattern:
# 6&lt;math&gt;\tfrac{98}{100}&lt;/math&gt; = 6.98
# 2&lt;math&gt;\tfrac{56}{100}&lt;/math&gt; = 2.56
# 9&lt;math&gt;\tfrac{27}{100}&lt;/math&gt; = 9.27
# 5&lt;math&gt;\tfrac{83}{100}&lt;/math&gt; = 5.83
You see? We simply put the mixed number in front of the dot, and with the numerator, we slap that behind the dot! Throw out the 100, it's not important when building your decimal. 

Decimals are all about place value, the value of a number in a specific place in a number. So, when we have &lt;math&gt;6.72&lt;/math&gt;, the &lt;math&gt;6&lt;/math&gt; is in the Ones place. Now, let's throw &lt;math&gt;9&lt;/math&gt; in the tens place, which is 10 times bigger than the Ones place: &lt;math&gt;96.72&lt;/math&gt;. But... that's doesn't seem enough, does it? Let's throw in a &lt;math&gt;6, 2, 8&lt;/math&gt; and a &lt;math&gt;3&lt;/math&gt; in there! And now, we have: &lt;math&gt;628,396.72&lt;/math&gt;.

Woah! That's a pretty big number, but we can easily break this number down to it's place value. Let's do it!

So, our number, &lt;math&gt;628,396.72&lt;/math&gt;, is the number we need to break down. Let's start from the decimal point, and move left:
* The number &lt;math&gt;6&lt;/math&gt; is in the Ones place. '''x10'''
* The number &lt;math&gt;9&lt;/math&gt; is in the Tens place. '''x10'''
* The number &lt;math&gt;3&lt;/math&gt; is in the Hundreds place. '''x10'''
* The number &lt;math&gt;8&lt;/math&gt; is in the Thousands place. '''x10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten thousands place. '''x10'''
* The number &lt;math&gt;6&lt;/math&gt; is in the Hundred Thousands place.

Now we have broken up the numbers left of the decimal--What about the numbers on the ''right''? Let's throw in a &lt;math&gt;5, 2, 4&lt;/math&gt; and a &lt;math&gt;7&lt;/math&gt;. Now, we have &lt;math&gt;628,396.725,247&lt;/math&gt;. Let's break this number up like we did above.

So, our number, &lt;math&gt;628,396.725,247&lt;/math&gt;, is the number we need to break down. This time, we need to start on the decimal point, and move ''right'':
* The number &lt;math&gt;7&lt;/math&gt; is in the Tenths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Hundredths place. '''x-10'''
* The number &lt;math&gt;5&lt;/math&gt; is in the Thousandths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten Thousandths place. '''x-10'''
* The number &lt;math&gt;4&lt;/math&gt; is in the Hundred Thousandths place. '''x-10'''
* The number &lt;math&gt;7&lt;/math&gt; is in the Millionths place. 

We have just now gone over the importance of Place Value in the Decimal World. Now, we will go into how to work with decimals, in the Decimal World!

See also: http://www.shmoop.com/fractions-decimals/place-value-naming-decimals.html

==== Adding/Subtracting Decimals ====
To add decimals, in addition column-style, put the decimals in its place with the decimals lined up. Then simply add on. So, for &lt;math&gt;1.5&lt;/math&gt; + &lt;math&gt;2.5&lt;/math&gt; we'd line up the decimal points. But, if we had a problem like &lt;math&gt;1.15&lt;/math&gt; + &lt;math&gt;2.0&lt;/math&gt;, we'd add a &lt;math&gt;0&lt;/math&gt; after the &lt;math&gt;0&lt;/math&gt; that is behind the decimal. Adding a zero to a place in a decimal means &quot;no value&quot;. So &lt;math&gt;10&lt;/math&gt; basically means no ones, and &lt;math&gt;100&lt;/math&gt;, means no ones or hundreds. Same things goes for subtracting as well folks.
=====Sample problems for ''adding/subtracting decimals''=====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
6.8 - 2.5 = { 4.3_6 }

{
|type=&quot;{}&quot;}
3.4 + 5.6 = { 9_6 }

{
|type=&quot;{}&quot;}
9 + 4.50 = { 13.5_6 }

{
|type=&quot;{}&quot;}
41.89 + 25.00 = { 66.89_6 }

{
|type=&quot;{}&quot;}
9.01 + 3.089 = { 12.099_6 }

{
|type=&quot;{}&quot;}
10.90 + 11.1 = { 22_6 }

{
|type=&quot;{}&quot;}
9.5 + 3.44 = { 12.94_6 }

{
|type=&quot;{}&quot; coef=&quot;2.5&quot;}
9.00 x 2.00 = { 18_6 } 
&lt;/quiz&gt;

==== Multiplying Decimals ====
[[File:9.82x5.73 multiplication image.svg|thumb|A visual representation of the multiplication example]]
Multiplying decimals isn't as hard as it really seems to be. So, we have &lt;math&gt;9.83&lt;/math&gt; × &lt;math&gt;5.73&lt;/math&gt;. For most people, column multiplication is a lot easier than side-by-side multiplication. That being mentioned, let us column these numbers:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;× &lt;math&gt;5.73&lt;/math&gt;
-------

Now that we have our problem, we should simply ignore the decimal points and just multiply as usual, so you should get this answer once you are done with that (remember to add a zero (and grow with zeros in each line) to each and every line of addition):

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt;   &lt;math&gt;68810&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt; &lt;math&gt;491500
&lt;/math&gt;
-------

With the simple usage of addition, we should get:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;   &lt;math&gt;+&lt;/math&gt;  &lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+
&lt;/math&gt; &lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;563259&lt;/math&gt;

Now, we need to bring back our handy dandy decimal point, but where? In &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;, there are FOUR numbers in these 2 numbers overall that are behind the decimal point (in each number, there are two numbers behind the decimal points). So, we have &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;. Now, that totals up to four numbers overall behind the decimal point. So in &lt;math&gt;563259&lt;/math&gt;, we need to move the decimal point four times (beginning from the right). So watch as follows:

&lt;math&gt;563259.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56325.9&lt;/math&gt;&lt;br&gt;
&lt;math&gt;5632.59&lt;/math&gt;&lt;br&gt;
&lt;math&gt;563.259&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56.3259&lt;/math&gt;

That simple. Now, review your work, your whole work should look like this:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;       × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;    &lt;math&gt;+&lt;/math&gt;&lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+&lt;/math&gt;&lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;56.3259&lt;/math&gt;

==== Dividing Decimals ====
;Dividing a decimal by a whole number
If you want to divide a decimal by a whole number, you should divide the 2 numbers, omitting the decimal point. After you are done dividing, add the decimal point to the '''quotient''' (final product/answer at the top of the long division symbol). The decimal should be right above the decimal point in the '''dividend''' (number in the box/number that is being divided). It's quite easy and simple, as long as you know how to do long division and if you are still familiar with long division. 

Hey, this seems ''too'' easy--Let's figure out how to divide a decimal by a decimal!
;Dividing a decimal by a decimal
The trick to dividing a decimal by a decimal is to shift the decimal point as many times as it gets to a whole number, so follow along: &lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt;. Now, we simply move the decimal point as many times as we need to make the number we are going to use to divide 69.45 a whole number, so watch as followed:&lt;br&gt;
&lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;694.5&lt;/math&gt; ÷ &lt;math&gt;57.8&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;6945&lt;/math&gt;. ÷ &lt;math&gt;578&lt;/math&gt;.


Now that we have finally got our dividend a whole number (and now our first number that we are going to divide), we can go ahead and divide normally (using long division). In the end, &lt;math&gt;69.45&lt;/math&gt; divided by &lt;math&gt;5.78&lt;/math&gt; should get you &lt;math&gt;12.0155709&lt;/math&gt;!

A pretty simple one we could go is &lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;, here, we simply move our dots like so:&lt;br&gt;
&lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;04.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;4&lt;/math&gt;&lt;br&gt;
Then, we can simply divide, heck... we don't even need to do long division! The answer should pop in your head, which is &lt;math&gt;16&lt;/math&gt;.

{{notice|If you would like to take the quiz on Decimals, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Decimals Quiz]]'''}}

===Percentages===
A good definition of &quot;percent&quot; is a fraction in which the denominator is the number &lt;math&gt;100&lt;/math&gt;. For example, the numbers &lt;math&gt;59%&lt;/math&gt;, &lt;math&gt;63%&lt;/math&gt;, &lt;math&gt;91%&lt;/math&gt;, and &lt;math&gt;85%&lt;/math&gt;, are the same as just saying &lt;math&gt;\tfrac{59}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{63}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{91}{100}&lt;/math&gt;, and &lt;math&gt;\tfrac{85}{100}&lt;/math&gt;. You could also say 59 out of 100 parts, 63 out of 100 parts, 91 out of 100 parts, and 85 out of 100 parts.
====Converting Percentages====
Now that we got the basis of percentages and how they operate, we should look into changing percentages.
===== Percentage → Decimal =====
Let's look in turning a percentage into a decimal point first. It's very simple. Let's say you have &lt;math&gt;\tfrac{9}{100}&lt;/math&gt;, which, in percentage form, is &lt;math&gt;9%&lt;/math&gt;. So, we have 9%. Now, we want to change it to a decimal (I don't know, think of a reason). We simply convert the percentage symbol into a decimal point, so like this: &lt;math&gt;9.&lt;/math&gt;. Now, we have &lt;math&gt;9.&lt;/math&gt;, so then we move the decimal number two places to the left, like so: &lt;math&gt;9.&lt;/math&gt; → &lt;math&gt;.9&lt;/math&gt; → &lt;math&gt;.09&lt;/math&gt;. So now, we have &lt;math&gt;0.09&lt;/math&gt;. We added the 2 zeros in because there is no value in the tenths place, and because &lt;math&gt;.09&lt;/math&gt; does not look quite right. Looks a bit off.
===== Samples problems for ''converting percentages to decimals'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
59% = { 0.59_5 }

{
|type=&quot;{}&quot;}
63% = { 0.63_5 }

{
|type=&quot;{}&quot;}
91% = { 0.91_5 }

{
|type=&quot;{}&quot;}
85% = { 0.85_5 }

{
|type=&quot;{}&quot;}
9% = { 0.09_5 }

{
|type=&quot;{}&quot;}
9834% = { 98.34_5 }

{
|type=&quot;{}&quot;}
20% = { 0.2_5 }

{
|type=&quot;{}&quot;}
4% = { 0.04_5 }

{
|type=&quot;{}&quot;}
7.6% = { 0.076_5 }

{
|type=&quot;{}&quot;}
6% = { 0.06_5 }
&lt;/quiz&gt;
===== Decimal → Percentage =====
Now to convert a decimal into percentage we essentially do the complete opposite. We have &lt;math&gt;98.34&lt;/math&gt;. We need this to be a percentage (easier to read). Move the decimal point two places to the right. So, watch: &lt;math&gt;98.34&lt;/math&gt; → &lt;math&gt;983.4&lt;/math&gt; → &lt;math&gt;9834.&lt;/math&gt; --Now, we have &lt;math&gt;9834.&lt;/math&gt;, but the decimal point, since it's now a percentage, should not be there, but instead, a percentage should talk the decimal point's place. Now, we have our final result of &lt;math&gt;9834%&lt;/math&gt;.

==== Finding percent of a number ====
[[File:Universität Bonn.jpg|thumb|right|Would this be the fictional university these students were trying to get accepted to?]]
So, 95 students applied to a university (the fictional [[User:Atcovi/Mustafa Einhoonansebadoi University|Mustafa Einhoonansebadoi University]], for example), and only 20% of the students made it. 20%? What? With this in mind, we want to find &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt;. We take the percentage, &lt;math&gt;20%&lt;/math&gt;, and divide it by &lt;math&gt;100&lt;/math&gt;. So we get &lt;math&gt;20/100&lt;/math&gt; = &lt;math&gt;.2&lt;/math&gt;. Then, we multiply &lt;math&gt;.2&lt;/math&gt; by &lt;math&gt;95&lt;/math&gt;, in which we get &lt;math&gt;19&lt;/math&gt;. So &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt; is &lt;math&gt;19&lt;/math&gt;. If that didn't make sense .2 x 95 is 2 x 95 = 190 but because we multipled 2 by ten in the tenths place to the ones place we -10 off of the number to reverse what we just did.  Therefore, only 19 students out of 95 students made it into the fictional Mustafa Einhoonansebadoi University.</text>
      <sha1>st9lt2hrezspo7axxbru4cw03dxha8p</sha1>
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  </page>
  <page>
    <title>OpenStax Nutrition for Nurses</title>
    <ns>0</ns>
    <id>330717</id>
    <revision>
      <id>2819243</id>
      <timestamp>2026-07-24T11:55:43Z</timestamp>
      <contributor>
        <username>Andy?yes</username>
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      <text bytes="618" sha1="qbuv1rl3ssu2rvhx9hfif9g6fuo3htr" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; Nutrition for Nurses ==

== Summary ==
''Nutrition for Nurses'' is structured to support the flexible integration of nutrition content across both system-based and nursing competency-based curricula. It can be used whether nutrition is taught as a standalone course or part of another nursing course. The table of contents for ''Nutrition for Nurses'' presents content in 20 chapters, organized into 9 thematic units.

* OpenStax Nutrition for Nurses (original content). Available as pdf or web view.
* OpenStax Nutrition for Nurses audiobook. Available as audio textbook.</text>
      <sha1>qbuv1rl3ssu2rvhx9hfif9g6fuo3htr</sha1>
    </revision>
    <revision>
      <id>2819244</id>
      <parentid>2819243</parentid>
      <timestamp>2026-07-24T11:57:05Z</timestamp>
      <contributor>
        <username>Andy?yes</username>
        <id>3006471</id>
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      <origin>2819244</origin>
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      <format>text/x-wiki</format>
      <text bytes="1467" sha1="91dfzw7jwbt37nhopx9qupu5n0r1bng" xml:space="preserve">See also [[OpenStax]]

== &lt;big&gt;OpenStax&lt;/big&gt; Nutrition for Nurses ==

== Summary ==
''Nutrition for Nurses'' is structured to support the flexible integration of nutrition content across both system-based and nursing competency-based curricula. It can be used whether nutrition is taught as a standalone course or part of another nursing course. The table of contents for ''Nutrition for Nurses'' presents content in 20 chapters, organized into 9 thematic units.

The text emphasizes evidence-based practice and holistic assessment to facilitate the integration of nutritional awareness for pre-licensure nursing students in the provision of client-centered care. ''Nutrition for Nurses'' helps students develop sound clinical judgment as well as a deep understanding of the impact of nutrition on body systems across the lifespan.

Written and reviewed by highly experienced faculty, ''Nutrition for Nurses'' includes a detailed narrative, extensive features and learning resources, and ample student support. The presentation utilizes concepts promoting the development of clinical judgment by building upon the systematic model developed by the National Council of State Boards of Nursing (NCSBN).

* [https://openstax.org/details/books/nutrition OpenStax Nutrition for Nurses] (original content). Available as pdf or web view.
* [https://audileo.com/audiobooks/openstax/nutrition-for-nurses/ OpenStax Nutrition for Nurses audiobook]. Available as audio textbook.</text>
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